id	sid	tid	token	lemma	pos
ejpam-4412	1	1	european	european	PROPN
ejpam-4412	1	2	journal	journal	PROPN
ejpam-4412	1	3	of	of	ADP
ejpam-4412	1	4	pure	pure	ADJ
ejpam-4412	1	5	and	and	CCONJ
ejpam-4412	1	6	applied	apply	VERB
ejpam-4412	1	7	mathematics	mathematic	NOUN
ejpam-4412	1	8	vol	vol	NOUN
ejpam-4412	1	9	.	.	PROPN
ejpam-4412	2	1	15	15	NUM
ejpam-4412	2	2	,	,	PUNCT
ejpam-4412	2	3	no	no	INTJ
ejpam-4412	2	4	.	.	NOUN
ejpam-4412	2	5	3	3	NUM
ejpam-4412	2	6	,	,	PUNCT
ejpam-4412	2	7	2022	2022	NUM
ejpam-4412	2	8	,	,	PUNCT
ejpam-4412	2	9	1067	1067	NUM
ejpam-4412	2	10	-	-	SYM
ejpam-4412	2	11	1089	1089	NUM
ejpam-4412	2	12	issn	issn	PROPN
ejpam-4412	2	13	1307	1307	NUM
ejpam-4412	2	14	-	-	SYM
ejpam-4412	2	15	5543	5543	NUM
ejpam-4412	2	16	–	–	PUNCT
ejpam-4412	3	1	ejpam.com	ejpam.com	X
ejpam-4412	3	2	published	publish	VERB
ejpam-4412	3	3	by	by	ADP
ejpam-4412	3	4	new	new	PROPN
ejpam-4412	3	5	york	york	PROPN
ejpam-4412	3	6	business	business	PROPN
ejpam-4412	3	7	global	global	PROPN
ejpam-4412	3	8	the	the	DET
ejpam-4412	3	9	fuglede	fuglede	PROPN
ejpam-4412	3	10	-	-	PUNCT
ejpam-4412	3	11	putnam	putnam	NOUN
ejpam-4412	3	12	theorem	theorem	NOUN
ejpam-4412	3	13	and	and	CCONJ
ejpam-4412	3	14	quasinormality	quasinormality	NOUN
ejpam-4412	3	15	for	for	ADP
ejpam-4412	3	16	class	class	NOUN
ejpam-4412	3	17	p	p	NOUN
ejpam-4412	3	18	-	-	PUNCT
ejpam-4412	3	19	wa(s	wa(s	NUM
ejpam-4412	3	20	,	,	PUNCT
ejpam-4412	3	21	t	t	PROPN
ejpam-4412	3	22	)	)	PUNCT
ejpam-4412	3	23	operators	operator	NOUN
ejpam-4412	3	24	m.h.m.rashid1,∗	m.h.m.rashid1,∗	PROPN
ejpam-4412	3	25	,	,	PUNCT
ejpam-4412	3	26	n.	n.	PROPN
ejpam-4412	3	27	h.	h.	PROPN
ejpam-4412	4	1	altaweel2	altaweel2	PROPN
ejpam-4412	5	1	1	1	NUM
ejpam-4412	6	1	department	department	NOUN
ejpam-4412	6	2	of	of	ADP
ejpam-4412	6	3	mathematics	mathematic	NOUN
ejpam-4412	6	4	,	,	PUNCT
ejpam-4412	6	5	faculty	faculty	NOUN
ejpam-4412	6	6	of	of	ADP
ejpam-4412	6	7	science	science	PROPN
ejpam-4412	6	8	p.o	p.o	PROPN
ejpam-4412	6	9	.	.	PROPN
ejpam-4412	6	10	box(7	box(7	PROPN
ejpam-4412	6	11	)	)	PUNCT
ejpam-4412	6	12	,	,	PUNCT
ejpam-4412	6	13	mu’tah	mu’tah	PROPN
ejpam-4412	6	14	university	university	PROPN
ejpam-4412	6	15	,	,	PUNCT
ejpam-4412	6	16	al	al	PROPN
ejpam-4412	6	17	-	-	PUNCT
ejpam-4412	6	18	karakjordan	karakjordan	PROPN
ejpam-4412	6	19	2	2	NUM
ejpam-4412	6	20	department	department	NOUN
ejpam-4412	6	21	of	of	ADP
ejpam-4412	6	22	mathematics	mathematic	NOUN
ejpam-4412	6	23	-	-	PUNCT
ejpam-4412	6	24	faculty	faculty	NOUN
ejpam-4412	6	25	of	of	ADP
ejpam-4412	6	26	science	science	NOUN
ejpam-4412	6	27	,	,	PUNCT
ejpam-4412	6	28	university	university	NOUN
ejpam-4412	6	29	of	of	ADP
ejpam-4412	6	30	tabuk	tabuk	PROPN
ejpam-4412	6	31	,	,	PUNCT
ejpam-4412	6	32	p.o.box	p.o.box	ADP
ejpam-4412	6	33	741tabuk	741tabuk	PROPN
ejpam-4412	6	34	71491	71491	NUM
ejpam-4412	6	35	,	,	PUNCT
ejpam-4412	6	36	saudi	saudi	PROPN
ejpam-4412	6	37	arabia	arabia	PROPN
ejpam-4412	6	38	abstract	abstract	NOUN
ejpam-4412	6	39	.	.	PUNCT
ejpam-4412	7	1	in	in	ADP
ejpam-4412	7	2	this	this	DET
ejpam-4412	7	3	work	work	NOUN
ejpam-4412	7	4	,	,	PUNCT
ejpam-4412	7	5	we	we	PRON
ejpam-4412	7	6	demonstrate	demonstrate	VERB
ejpam-4412	7	7	that	that	SCONJ
ejpam-4412	7	8	(	(	PUNCT
ejpam-4412	7	9	i	i	NOUN
ejpam-4412	7	10	)	)	PUNCT
ejpam-4412	7	11	if	if	SCONJ
ejpam-4412	7	12	t	t	PROPN
ejpam-4412	7	13	is	be	AUX
ejpam-4412	7	14	a	a	DET
ejpam-4412	7	15	class	class	NOUN
ejpam-4412	7	16	p	p	NOUN
ejpam-4412	7	17	-	-	PUNCT
ejpam-4412	7	18	wa(s	wa(s	NUM
ejpam-4412	7	19	,	,	PUNCT
ejpam-4412	7	20	t	t	NOUN
ejpam-4412	7	21	)	)	PUNCT
ejpam-4412	7	22	operator	operator	NOUN
ejpam-4412	7	23	and	and	CCONJ
ejpam-4412	7	24	t	t	PROPN
ejpam-4412	7	25	(	(	PUNCT
ejpam-4412	7	26	s	s	PROPN
ejpam-4412	7	27	,	,	PUNCT
ejpam-4412	7	28	t	t	PROPN
ejpam-4412	7	29	)	)	PUNCT
ejpam-4412	7	30	is	be	AUX
ejpam-4412	7	31	quasinormal	quasinormal	ADJ
ejpam-4412	7	32	(	(	PUNCT
ejpam-4412	7	33	resp	resp	NOUN
ejpam-4412	7	34	.	.	PUNCT
ejpam-4412	7	35	,	,	PUNCT
ejpam-4412	7	36	normal	normal	ADJ
ejpam-4412	7	37	)	)	PUNCT
ejpam-4412	7	38	,	,	PUNCT
ejpam-4412	7	39	then	then	ADV
ejpam-4412	7	40	t	t	PROPN
ejpam-4412	7	41	is	be	AUX
ejpam-4412	7	42	also	also	ADV
ejpam-4412	7	43	quasinormal	quasinormal	ADJ
ejpam-4412	7	44	(	(	PUNCT
ejpam-4412	7	45	resp	resp	NOUN
ejpam-4412	7	46	.	.	PUNCT
ejpam-4412	7	47	,	,	PUNCT
ejpam-4412	7	48	normal	normal	ADJ
ejpam-4412	7	49	)	)	PUNCT
ejpam-4412	7	50	(	(	PUNCT
ejpam-4412	7	51	ii	ii	NOUN
ejpam-4412	7	52	)	)	PUNCT
ejpam-4412	7	53	if	if	SCONJ
ejpam-4412	7	54	t	t	PROPN
ejpam-4412	7	55	and	and	CCONJ
ejpam-4412	7	56	t∗	t∗	PROPN
ejpam-4412	7	57	are	be	AUX
ejpam-4412	7	58	class	class	NOUN
ejpam-4412	7	59	p	p	NOUN
ejpam-4412	7	60	-	-	PUNCT
ejpam-4412	7	61	wa(s	wa(s	NUM
ejpam-4412	7	62	,	,	PUNCT
ejpam-4412	7	63	t	t	NOUN
ejpam-4412	7	64	)	)	PUNCT
ejpam-4412	7	65	operators	operator	NOUN
ejpam-4412	7	66	,	,	PUNCT
ejpam-4412	7	67	then	then	ADV
ejpam-4412	7	68	t	t	PROPN
ejpam-4412	7	69	is	be	AUX
ejpam-4412	7	70	normal	normal	ADJ
ejpam-4412	7	71	;	;	PUNCT
ejpam-4412	7	72	(	(	PUNCT
ejpam-4412	7	73	iii	iii	X
ejpam-4412	7	74	)	)	PUNCT
ejpam-4412	7	75	the	the	DET
ejpam-4412	7	76	normal	normal	ADJ
ejpam-4412	7	77	portions	portion	NOUN
ejpam-4412	7	78	of	of	ADP
ejpam-4412	7	79	quasisimilar	quasisimilar	ADJ
ejpam-4412	7	80	class	class	NOUN
ejpam-4412	7	81	pwa(s	pwa(s	PROPN
ejpam-4412	7	82	,	,	PUNCT
ejpam-4412	7	83	t	t	PROPN
ejpam-4412	7	84	)	)	PUNCT
ejpam-4412	7	85	operators	operator	NOUN
ejpam-4412	7	86	are	be	AUX
ejpam-4412	7	87	unitarily	unitarily	ADV
ejpam-4412	7	88	equivalent	equivalent	ADJ
ejpam-4412	7	89	;	;	PUNCT
ejpam-4412	7	90	and	and	CCONJ
ejpam-4412	7	91	(	(	PUNCT
ejpam-4412	7	92	iv	iv	X
ejpam-4412	7	93	)	)	PUNCT
ejpam-4412	7	94	fuglede	fuglede	PROPN
ejpam-4412	7	95	-	-	PUNCT
ejpam-4412	7	96	putnam	putnam	NOUN
ejpam-4412	7	97	type	type	NOUN
ejpam-4412	7	98	theorem	theorem	NOUN
ejpam-4412	7	99	holds	hold	VERB
ejpam-4412	7	100	for	for	ADP
ejpam-4412	7	101	a	a	DET
ejpam-4412	7	102	class	class	NOUN
ejpam-4412	7	103	p	p	NOUN
ejpam-4412	7	104	-	-	PUNCT
ejpam-4412	7	105	wa(s	wa(s	NUM
ejpam-4412	7	106	,	,	PUNCT
ejpam-4412	7	107	t	t	NOUN
ejpam-4412	7	108	)	)	PUNCT
ejpam-4412	7	109	operator	operator	NOUN
ejpam-4412	7	110	t	t	NOUN
ejpam-4412	7	111	for	for	ADP
ejpam-4412	7	112	0	0	NUM
ejpam-4412	7	113	<	<	X
ejpam-4412	7	114	s	s	PROPN
ejpam-4412	7	115	,	,	PUNCT
ejpam-4412	7	116	t	t	PROPN
ejpam-4412	7	117	,	,	PUNCT
ejpam-4412	7	118	s	s	PART
ejpam-4412	7	119	+	+	NUM
ejpam-4412	7	120	t	t	X
ejpam-4412	7	121	=	=	SYM
ejpam-4412	7	122	1	1	NUM
ejpam-4412	7	123	and	and	CCONJ
ejpam-4412	7	124	0	0	NUM
ejpam-4412	8	1	<	<	X
ejpam-4412	8	2	p	p	X
ejpam-4412	8	3	≤	≤	NOUN
ejpam-4412	8	4	1	1	NUM
ejpam-4412	8	5	if	if	SCONJ
ejpam-4412	8	6	t	t	PROPN
ejpam-4412	8	7	satisfies	satisfy	VERB
ejpam-4412	8	8	a	a	DET
ejpam-4412	8	9	kernel	kernel	NOUN
ejpam-4412	8	10	condition	condition	NOUN
ejpam-4412	8	11	ker(t	ker(t	NOUN
ejpam-4412	8	12	)	)	PUNCT
ejpam-4412	8	13	⊂	⊂	PROPN
ejpam-4412	8	14	ker(t	ker(t	NOUN
ejpam-4412	8	15	∗	∗	NOUN
ejpam-4412	8	16	)	)	PUNCT
ejpam-4412	8	17	.	.	PUNCT
ejpam-4412	9	1	2020	2020	NUM
ejpam-4412	9	2	mathematics	mathematic	NOUN
ejpam-4412	9	3	subject	subject	NOUN
ejpam-4412	9	4	classifications	classification	NOUN
ejpam-4412	9	5	:	:	PUNCT
ejpam-4412	9	6	47a10	47a10	NUM
ejpam-4412	9	7	,	,	PUNCT
ejpam-4412	9	8	47a11	47a11	NUM
ejpam-4412	9	9	,	,	PUNCT
ejpam-4412	9	10	47b20	47b20	NUM
ejpam-4412	9	11	key	key	ADJ
ejpam-4412	9	12	words	word	NOUN
ejpam-4412	9	13	and	and	CCONJ
ejpam-4412	9	14	phrases	phrase	NOUN
ejpam-4412	9	15	:	:	PUNCT
ejpam-4412	9	16	quasinormal	quasinormal	ADJ
ejpam-4412	9	17	,	,	PUNCT
ejpam-4412	9	18	class	class	PROPN
ejpam-4412	9	19	a(s	a(s	PROPN
ejpam-4412	9	20	,	,	PUNCT
ejpam-4412	9	21	t	t	PROPN
ejpam-4412	9	22	)	)	PUNCT
ejpam-4412	9	23	operators	operator	NOUN
ejpam-4412	9	24	,	,	PUNCT
ejpam-4412	9	25	class	class	NOUN
ejpam-4412	9	26	p-(a(s	p-(a(s	NOUN
ejpam-4412	9	27	,	,	PUNCT
ejpam-4412	9	28	t	t	PROPN
ejpam-4412	9	29	)	)	PUNCT
ejpam-4412	9	30	operators	operator	NOUN
ejpam-4412	9	31	,	,	PUNCT
ejpam-4412	9	32	fuglede	fuglede	PROPN
ejpam-4412	9	33	-	-	PUNCT
ejpam-4412	9	34	putnam	putnam	NOUN
ejpam-4412	9	35	theorem	theorem	NOUN
ejpam-4412	9	36	1	1	NUM
ejpam-4412	9	37	.	.	PUNCT
ejpam-4412	10	1	introduction	introduction	NOUN
ejpam-4412	10	2	on	on	ADP
ejpam-4412	10	3	a	a	DET
ejpam-4412	10	4	complex	complex	ADJ
ejpam-4412	10	5	hilbert	hilbert	NOUN
ejpam-4412	10	6	space	space	NOUN
ejpam-4412	10	7	h	h	NOUN
ejpam-4412	10	8	,	,	PUNCT
ejpam-4412	10	9	let	let	VERB
ejpam-4412	10	10	b(h	b(h	NOUN
ejpam-4412	10	11	)	)	PUNCT
ejpam-4412	10	12	be	be	VERB
ejpam-4412	10	13	the	the	DET
ejpam-4412	10	14	algebra	algebra	NOUN
ejpam-4412	10	15	of	of	ADP
ejpam-4412	10	16	all	all	DET
ejpam-4412	10	17	bounded	bounded	ADJ
ejpam-4412	10	18	linear	linear	PROPN
ejpam-4412	10	19	operators	operator	NOUN
ejpam-4412	10	20	.	.	PUNCT
ejpam-4412	11	1	aluthge	aluthge	PROPN
ejpam-4412	12	1	[	[	X
ejpam-4412	12	2	2	2	NUM
ejpam-4412	12	3	]	]	PUNCT
ejpam-4412	12	4	investigated	investigate	VERB
ejpam-4412	12	5	the	the	DET
ejpam-4412	12	6	p	p	PROPN
ejpam-4412	12	7	-	-	PUNCT
ejpam-4412	12	8	hyponormal	hyponormal	ADJ
ejpam-4412	12	9	operator	operator	NOUN
ejpam-4412	12	10	t	t	PROPN
ejpam-4412	12	11	,	,	PUNCT
ejpam-4412	12	12	which	which	PRON
ejpam-4412	12	13	is	be	AUX
ejpam-4412	12	14	defined	define	VERB
ejpam-4412	12	15	as	as	ADP
ejpam-4412	12	16	(	(	PUNCT
ejpam-4412	12	17	t	t	NOUN
ejpam-4412	12	18	∗t	∗t	PROPN
ejpam-4412	12	19	)	)	PUNCT
ejpam-4412	12	20	p	p	NOUN
ejpam-4412	12	21	≥	≥	X
ejpam-4412	12	22	(	(	PUNCT
ejpam-4412	12	23	tt	tt	PROPN
ejpam-4412	12	24	∗)p	∗)p	PROPN
ejpam-4412	12	25	with	with	ADP
ejpam-4412	12	26	0	0	NUM
ejpam-4412	12	27	≤	≤	NOUN
ejpam-4412	12	28	p	p	NOUN
ejpam-4412	12	29	≤	≤	NUM
ejpam-4412	12	30	1	1	NUM
ejpam-4412	12	31	using	use	VERB
ejpam-4412	12	32	the	the	DET
ejpam-4412	12	33	furuta	furuta	ADJ
ejpam-4412	12	34	inequality	inequality	NOUN
ejpam-4412	12	35	[	[	X
ejpam-4412	12	36	14	14	NUM
ejpam-4412	12	37	]	]	PUNCT
ejpam-4412	12	38	.	.	PUNCT
ejpam-4412	13	1	when	when	SCONJ
ejpam-4412	13	2	p	p	NOUN
ejpam-4412	13	3	=	=	NOUN
ejpam-4412	13	4	1	1	NUM
ejpam-4412	13	5	,	,	PUNCT
ejpam-4412	13	6	t	t	PROPN
ejpam-4412	13	7	is	be	AUX
ejpam-4412	13	8	said	say	VERB
ejpam-4412	13	9	to	to	PART
ejpam-4412	13	10	be	be	AUX
ejpam-4412	13	11	hyponormal	hyponormal	ADJ
ejpam-4412	13	12	.	.	PUNCT
ejpam-4412	14	1	as	as	ADP
ejpam-4412	14	2	a	a	DET
ejpam-4412	14	3	result	result	NOUN
ejpam-4412	14	4	,	,	PUNCT
ejpam-4412	14	5	p	p	X
ejpam-4412	14	6	-	-	PUNCT
ejpam-4412	14	7	hyponormality	hyponormality	NOUN
ejpam-4412	14	8	is	be	AUX
ejpam-4412	14	9	a	a	DET
ejpam-4412	14	10	broadening	broadening	NOUN
ejpam-4412	14	11	of	of	ADP
ejpam-4412	14	12	hyponormality	hyponormality	NOUN
ejpam-4412	14	13	.	.	PUNCT
ejpam-4412	15	1	following	follow	VERB
ejpam-4412	15	2	[	[	X
ejpam-4412	15	3	2	2	NUM
ejpam-4412	15	4	]	]	PUNCT
ejpam-4412	15	5	,	,	PUNCT
ejpam-4412	15	6	several	several	ADJ
ejpam-4412	15	7	authors	author	NOUN
ejpam-4412	15	8	are	be	AUX
ejpam-4412	15	9	looking	look	VERB
ejpam-4412	15	10	towards	towards	ADP
ejpam-4412	15	11	novel	novel	ADJ
ejpam-4412	15	12	hyponormal	hyponormal	ADJ
ejpam-4412	15	13	operator	operator	NOUN
ejpam-4412	15	14	generalizations	generalization	NOUN
ejpam-4412	15	15	.	.	PUNCT
ejpam-4412	16	1	it	it	PRON
ejpam-4412	16	2	is	be	AUX
ejpam-4412	16	3	known	know	VERB
ejpam-4412	16	4	that	that	SCONJ
ejpam-4412	16	5	p	p	PROPN
ejpam-4412	16	6	-	-	PUNCT
ejpam-4412	16	7	hyponormal	hyponormal	ADJ
ejpam-4412	16	8	operators	operator	NOUN
ejpam-4412	16	9	have	have	VERB
ejpam-4412	16	10	many	many	ADJ
ejpam-4412	16	11	interesting	interesting	ADJ
ejpam-4412	16	12	properties	property	NOUN
ejpam-4412	16	13	as	as	ADP
ejpam-4412	16	14	hyponormal	hyponormal	ADJ
ejpam-4412	16	15	operators	operator	NOUN
ejpam-4412	16	16	,	,	PUNCT
ejpam-4412	16	17	for	for	ADP
ejpam-4412	16	18	example	example	NOUN
ejpam-4412	16	19	,	,	PUNCT
ejpam-4412	16	20	putnam	putnam	PROPN
ejpam-4412	16	21	’s	’s	PART
ejpam-4412	16	22	inequality	inequality	NOUN
ejpam-4412	16	23	,	,	PUNCT
ejpam-4412	16	24	fuglede	fuglede	NOUN
ejpam-4412	16	25	-	-	PUNCT
ejpam-4412	16	26	putnam	putnam	NOUN
ejpam-4412	16	27	type	type	NOUN
ejpam-4412	16	28	theorem	theorem	PROPN
ejpam-4412	16	29	,	,	PUNCT
ejpam-4412	16	30	bishop	bishop	PROPN
ejpam-4412	16	31	’s	’s	PART
ejpam-4412	16	32	property	property	NOUN
ejpam-4412	16	33	(	(	PUNCT
ejpam-4412	16	34	β	β	NOUN
ejpam-4412	16	35	)	)	PUNCT
ejpam-4412	16	36	,	,	PUNCT
ejpam-4412	16	37	weyl	weyl	PROPN
ejpam-4412	16	38	’s	’s	PART
ejpam-4412	16	39	theorem	theorem	NOUN
ejpam-4412	16	40	and	and	CCONJ
ejpam-4412	16	41	polaroid	polaroid	NOUN
ejpam-4412	16	42	.	.	PUNCT
ejpam-4412	17	1	let	let	VERB
ejpam-4412	17	2	t	t	PROPN
ejpam-4412	17	3	∈	∈	PROPN
ejpam-4412	17	4	b(h	b(h	PROPN
ejpam-4412	17	5	)	)	PUNCT
ejpam-4412	17	6	and	and	CCONJ
ejpam-4412	17	7	|t	|t	VERB
ejpam-4412	18	1	|	|	ADV
ejpam-4412	18	2	=	=	SYM
ejpam-4412	18	3	(	(	PUNCT
ejpam-4412	18	4	t	t	PROPN
ejpam-4412	18	5	∗t	∗t	ADJ
ejpam-4412	18	6	)	)	PUNCT
ejpam-4412	18	7	1	1	NUM
ejpam-4412	18	8	2	2	NUM
ejpam-4412	18	9	.	.	PUNCT
ejpam-4412	19	1	by	by	ADP
ejpam-4412	19	2	taking	take	VERB
ejpam-4412	19	3	u	u	PRON
ejpam-4412	19	4	|t	|t	NOUN
ejpam-4412	19	5	|x	|x	NOUN
ejpam-4412	19	6	=	=	PUNCT
ejpam-4412	19	7	tx	tx	PROPN
ejpam-4412	19	8	for	for	ADP
ejpam-4412	19	9	x	x	PROPN
ejpam-4412	19	10	∈	∈	PROPN
ejpam-4412	19	11	h	h	NOUN
ejpam-4412	19	12	and	and	CCONJ
ejpam-4412	19	13	ux	ux	NOUN
ejpam-4412	20	1	=	=	NOUN
ejpam-4412	20	2	0	0	NUM
ejpam-4412	21	1	for	for	ADP
ejpam-4412	21	2	x	x	PROPN
ejpam-4412	21	3	∈	∈	PROPN
ejpam-4412	21	4	ker	ker	PROPN
ejpam-4412	21	5	|t	|t	VERB
ejpam-4412	22	1	|	|	ADV
ejpam-4412	22	2	,	,	PUNCT
ejpam-4412	22	3	t	t	PROPN
ejpam-4412	22	4	has	have	VERB
ejpam-4412	22	5	a	a	DET
ejpam-4412	22	6	unique	unique	ADJ
ejpam-4412	22	7	polar	polar	ADJ
ejpam-4412	22	8	decomposition	decomposition	NOUN
ejpam-4412	22	9	t	t	NOUN
ejpam-4412	22	10	=	=	SYM
ejpam-4412	22	11	u	u	NOUN
ejpam-4412	22	12	|t	|t	VERB
ejpam-4412	22	13	|	|	ADV
ejpam-4412	22	14	with	with	ADP
ejpam-4412	22	15	condition	condition	NOUN
ejpam-4412	22	16	keru	keru	NOUN
ejpam-4412	22	17	=	=	PROPN
ejpam-4412	22	18	ker	ker	PROPN
ejpam-4412	22	19	|t	|t	VERB
ejpam-4412	23	1	|	|	INTJ
ejpam-4412	23	2	.	.	PUNCT
ejpam-4412	24	1	we	we	PRON
ejpam-4412	24	2	say	say	VERB
ejpam-4412	24	3	that	that	DET
ejpam-4412	24	4	t	t	NOUN
ejpam-4412	24	5	=	=	SYM
ejpam-4412	24	6	u	u	SYM
ejpam-4412	24	7	|t	|t	NOUN
ejpam-4412	24	8	|	|	ADV
ejpam-4412	24	9	is	be	AUX
ejpam-4412	24	10	the	the	DET
ejpam-4412	24	11	polar	polar	ADJ
ejpam-4412	24	12	decomposition	decomposition	NOUN
ejpam-4412	24	13	of	of	ADP
ejpam-4412	24	14	t	t	PROPN
ejpam-4412	24	15	.	.	PUNCT
ejpam-4412	25	1	in	in	ADP
ejpam-4412	25	2	[	[	X
ejpam-4412	25	3	2	2	NUM
ejpam-4412	25	4	]	]	PUNCT
ejpam-4412	25	5	,	,	PUNCT
ejpam-4412	25	6	aluthge	aluthge	PROPN
ejpam-4412	25	7	extended	extend	VERB
ejpam-4412	25	8	the	the	DET
ejpam-4412	25	9	class	class	NOUN
ejpam-4412	25	10	of	of	ADP
ejpam-4412	25	11	hyponormal	hyponormal	ADJ
ejpam-4412	25	12	operators	operator	NOUN
ejpam-4412	25	13	by	by	ADP
ejpam-4412	25	14	introducing	introduce	VERB
ejpam-4412	25	15	p	p	PROPN
ejpam-4412	25	16	-	-	PUNCT
ejpam-4412	25	17	hyponormal	hyponormal	ADJ
ejpam-4412	25	18	operators	operator	NOUN
ejpam-4412	25	19	and	and	CCONJ
ejpam-4412	25	20	obtained	obtain	VERB
ejpam-4412	25	21	some	some	DET
ejpam-4412	25	22	properties	property	NOUN
ejpam-4412	25	23	with	with	ADP
ejpam-4412	25	24	the	the	DET
ejpam-4412	25	25	help	help	NOUN
ejpam-4412	25	26	of	of	ADP
ejpam-4412	25	27	the	the	DET
ejpam-4412	25	28	transformation	transformation	NOUN
ejpam-4412	25	29	∗corresponding	∗corresponde	VERB
ejpam-4412	25	30	author	author	NOUN
ejpam-4412	25	31	.	.	PUNCT
ejpam-4412	26	1	doi	doi	NOUN
ejpam-4412	26	2	:	:	PUNCT
ejpam-4412	26	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4412	https://doi.org/10.29020/nybg.ejpam.v15i3.4412	ADP
ejpam-4412	26	4	email	email	NOUN
ejpam-4412	26	5	addresses	address	VERB
ejpam-4412	26	6	:	:	PUNCT
ejpam-4412	26	7	malik	malik	PROPN
ejpam-4412	26	8	okasha@yahoo.com	okasha@yahoo.com	X
ejpam-4412	27	1	(	(	PUNCT
ejpam-4412	27	2	m.h.m	m.h.m	PROPN
ejpam-4412	27	3	.	.	PROPN
ejpam-4412	27	4	rashid	rashid	PROPN
ejpam-4412	27	5	)	)	PUNCT
ejpam-4412	27	6	,	,	PUNCT
ejpam-4412	27	7	naltawil@ut.edu.sa	naltawil@ut.edu.sa	PROPN
ejpam-4412	27	8	(	(	PUNCT
ejpam-4412	27	9	n.	n.	PROPN
ejpam-4412	27	10	h.	h.	PROPN
ejpam-4412	27	11	altaweel	altaweel	PROPN
ejpam-4412	27	12	)	)	PUNCT
ejpam-4412	27	13	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4412	27	14	1067	1067	NUM
ejpam-4412	28	1	©	©	PROPN
ejpam-4412	28	2	2022	2022	NUM
ejpam-4412	28	3	ejpam	ejpam	VERB
ejpam-4412	28	4	all	all	DET
ejpam-4412	28	5	rights	right	NOUN
ejpam-4412	28	6	reserved	reserve	VERB
ejpam-4412	28	7	.	.	PUNCT
ejpam-4412	29	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	29	2	,	,	PUNCT
ejpam-4412	29	3	n.	n.	NOUN
ejpam-4412	29	4	h.	h.	PROPN
ejpam-4412	29	5	altaweel	altaweel	PROPN
ejpam-4412	29	6	/	/	SYM
ejpam-4412	29	7	eur	eur	PROPN
ejpam-4412	29	8	.	.	PUNCT
ejpam-4412	30	1	j.	j.	PROPN
ejpam-4412	30	2	pure	pure	PROPN
ejpam-4412	30	3	appl	appl	PROPN
ejpam-4412	30	4	.	.	PROPN
ejpam-4412	30	5	math	math	PROPN
ejpam-4412	30	6	,	,	PUNCT
ejpam-4412	30	7	15	15	NUM
ejpam-4412	30	8	(	(	PUNCT
ejpam-4412	30	9	3	3	NUM
ejpam-4412	30	10	)	)	PUNCT
ejpam-4412	30	11	(	(	PUNCT
ejpam-4412	30	12	2022	2022	NUM
ejpam-4412	30	13	)	)	PUNCT
ejpam-4412	30	14	,	,	PUNCT
ejpam-4412	30	15	1067	1067	NUM
ejpam-4412	30	16	-	-	SYM
ejpam-4412	30	17	1089	1089	NUM
ejpam-4412	30	18	1068	1068	NUM
ejpam-4412	30	19	t	t	NOUN
ejpam-4412	30	20	(	(	PUNCT
ejpam-4412	30	21	12	12	NUM
ejpam-4412	30	22	,	,	PUNCT
ejpam-4412	30	23	1	1	NUM
ejpam-4412	30	24	2	2	NUM
ejpam-4412	30	25	)	)	PUNCT
ejpam-4412	30	26	=	=	VERB
ejpam-4412	31	1	|t	|t	VERB
ejpam-4412	32	1	|	|	ADV
ejpam-4412	32	2	1	1	NUM
ejpam-4412	32	3	2u	2u	NOUN
ejpam-4412	32	4	|t	|t	VERB
ejpam-4412	32	5	|	|	ADV
ejpam-4412	32	6	1	1	NUM
ejpam-4412	32	7	2	2	NUM
ejpam-4412	32	8	,	,	PUNCT
ejpam-4412	32	9	which	which	PRON
ejpam-4412	32	10	now	now	ADV
ejpam-4412	32	11	known	know	VERB
ejpam-4412	32	12	as	as	ADP
ejpam-4412	32	13	the	the	DET
ejpam-4412	32	14	aluthge	aluthge	ADJ
ejpam-4412	32	15	transform	transform	NOUN
ejpam-4412	32	16	.	.	PUNCT
ejpam-4412	33	1	the	the	DET
ejpam-4412	33	2	introduction	introduction	NOUN
ejpam-4412	33	3	of	of	ADP
ejpam-4412	33	4	these	these	DET
ejpam-4412	33	5	operators	operator	NOUN
ejpam-4412	33	6	by	by	ADP
ejpam-4412	33	7	aluthge	aluthge	PROPN
ejpam-4412	33	8	has	have	AUX
ejpam-4412	33	9	inspired	inspire	VERB
ejpam-4412	33	10	many	many	ADJ
ejpam-4412	33	11	researchers	researcher	NOUN
ejpam-4412	33	12	not	not	PART
ejpam-4412	33	13	only	only	ADV
ejpam-4412	33	14	to	to	PART
ejpam-4412	33	15	expose	expose	VERB
ejpam-4412	33	16	some	some	DET
ejpam-4412	33	17	important	important	ADJ
ejpam-4412	33	18	properties	property	NOUN
ejpam-4412	33	19	of	of	ADP
ejpam-4412	33	20	p	p	NOUN
ejpam-4412	33	21	-	-	PUNCT
ejpam-4412	33	22	hyponormal	hyponormal	ADJ
ejpam-4412	33	23	operators	operator	NOUN
ejpam-4412	33	24	but	but	CCONJ
ejpam-4412	33	25	also	also	ADV
ejpam-4412	33	26	to	to	PART
ejpam-4412	33	27	introduce	introduce	VERB
ejpam-4412	33	28	the	the	DET
ejpam-4412	33	29	number	number	NOUN
ejpam-4412	33	30	of	of	ADP
ejpam-4412	33	31	extensions	extension	NOUN
ejpam-4412	33	32	(	(	PUNCT
ejpam-4412	33	33	[	[	X
ejpam-4412	33	34	1	1	NUM
ejpam-4412	33	35	,	,	PUNCT
ejpam-4412	33	36	7	7	NUM
ejpam-4412	33	37	,	,	PUNCT
ejpam-4412	33	38	8	8	NUM
ejpam-4412	33	39	,	,	PUNCT
ejpam-4412	33	40	13	13	NUM
ejpam-4412	33	41	]	]	NUM
ejpam-4412	33	42	)	)	PUNCT
ejpam-4412	33	43	.	.	PUNCT
ejpam-4412	34	1	the	the	DET
ejpam-4412	34	2	aluthge	aluthge	ADJ
ejpam-4412	34	3	transform	transform	NOUN
ejpam-4412	34	4	,	,	PUNCT
ejpam-4412	34	5	and	and	CCONJ
ejpam-4412	34	6	more	more	ADV
ejpam-4412	34	7	broadly	broadly	ADV
ejpam-4412	34	8	,	,	PUNCT
ejpam-4412	34	9	the	the	DET
ejpam-4412	34	10	generalized	generalized	ADJ
ejpam-4412	34	11	aluthge	aluthge	ADJ
ejpam-4412	34	12	transform	transform	NOUN
ejpam-4412	34	13	defined	define	VERB
ejpam-4412	34	14	as	as	ADP
ejpam-4412	34	15	t	t	PROPN
ejpam-4412	34	16	(	(	PUNCT
ejpam-4412	34	17	s	s	PROPN
ejpam-4412	34	18	,	,	PUNCT
ejpam-4412	34	19	t	t	PROPN
ejpam-4412	34	20	)	)	PUNCT
ejpam-4412	34	21	=	=	PUNCT
ejpam-4412	34	22	|t	|t	PROPN
ejpam-4412	34	23	|su	|su	NOUN
ejpam-4412	34	24	|t	|t	VERB
ejpam-4412	34	25	|t	|t	VERB
ejpam-4412	34	26	with	with	ADP
ejpam-4412	34	27	s	s	PROPN
ejpam-4412	34	28	,	,	PUNCT
ejpam-4412	34	29	t	t	PROPN
ejpam-4412	34	30	>	>	X
ejpam-4412	34	31	0	0	PROPN
ejpam-4412	34	32	,	,	PUNCT
ejpam-4412	34	33	have	have	AUX
ejpam-4412	34	34	proven	prove	VERB
ejpam-4412	34	35	to	to	PART
ejpam-4412	34	36	be	be	AUX
ejpam-4412	34	37	useful	useful	ADJ
ejpam-4412	34	38	tools	tool	NOUN
ejpam-4412	34	39	in	in	ADP
ejpam-4412	34	40	this	this	DET
ejpam-4412	34	41	attempt	attempt	NOUN
ejpam-4412	34	42	.	.	PUNCT
ejpam-4412	35	1	the	the	DET
ejpam-4412	35	2	generalized	generalized	ADJ
ejpam-4412	35	3	aluthge	aluthge	ADJ
ejpam-4412	35	4	transform	transform	NOUN
ejpam-4412	35	5	is	be	AUX
ejpam-4412	35	6	used	use	VERB
ejpam-4412	35	7	to	to	PART
ejpam-4412	35	8	analyze	analyze	VERB
ejpam-4412	35	9	class	class	NOUN
ejpam-4412	35	10	p	p	NOUN
ejpam-4412	35	11	-	-	PUNCT
ejpam-4412	35	12	wa(s	wa(s	NUM
ejpam-4412	35	13	,	,	PUNCT
ejpam-4412	35	14	t	t	NOUN
ejpam-4412	35	15	)	)	PUNCT
ejpam-4412	35	16	operators	operator	NOUN
ejpam-4412	35	17	in	in	ADP
ejpam-4412	35	18	this	this	DET
ejpam-4412	35	19	article	article	NOUN
ejpam-4412	35	20	.	.	PUNCT
ejpam-4412	36	1	definition	definition	NOUN
ejpam-4412	36	2	1	1	NUM
ejpam-4412	36	3	.	.	PUNCT
ejpam-4412	37	1	let	let	AUX
ejpam-4412	37	2	t	t	NOUN
ejpam-4412	37	3	=	=	SYM
ejpam-4412	37	4	u	u	NOUN
ejpam-4412	37	5	|t	|t	VERB
ejpam-4412	37	6	|	|	ADV
ejpam-4412	37	7	be	be	AUX
ejpam-4412	37	8	the	the	DET
ejpam-4412	37	9	polar	polar	ADJ
ejpam-4412	37	10	decomposition	decomposition	NOUN
ejpam-4412	37	11	of	of	ADP
ejpam-4412	37	12	an	an	DET
ejpam-4412	37	13	operator	operator	NOUN
ejpam-4412	37	14	t	t	PROPN
ejpam-4412	37	15	∈	∈	PROPN
ejpam-4412	37	16	b(h	b(h	PROPN
ejpam-4412	37	17	)	)	PUNCT
ejpam-4412	37	18	.	.	PUNCT
ejpam-4412	38	1	then	then	ADV
ejpam-4412	38	2	the	the	DET
ejpam-4412	38	3	generalized	generalized	ADJ
ejpam-4412	38	4	aluthge	aluthge	ADJ
ejpam-4412	38	5	transform	transform	NOUN
ejpam-4412	38	6	t	t	PROPN
ejpam-4412	38	7	(	(	PUNCT
ejpam-4412	38	8	s	s	PROPN
ejpam-4412	38	9	,	,	PUNCT
ejpam-4412	38	10	t	t	PROPN
ejpam-4412	38	11	)	)	PUNCT
ejpam-4412	38	12	of	of	ADP
ejpam-4412	38	13	t	t	PROPN
ejpam-4412	38	14	is	be	AUX
ejpam-4412	38	15	defined	define	VERB
ejpam-4412	38	16	as	as	SCONJ
ejpam-4412	38	17	follows	follow	VERB
ejpam-4412	38	18	:	:	PUNCT
ejpam-4412	38	19	t	t	PROPN
ejpam-4412	38	20	(	(	PUNCT
ejpam-4412	38	21	s	s	PROPN
ejpam-4412	38	22	,	,	PUNCT
ejpam-4412	38	23	t	t	PROPN
ejpam-4412	38	24	)	)	PUNCT
ejpam-4412	38	25	=	=	PUNCT
ejpam-4412	39	1	|t	|t	PROPN
ejpam-4412	39	2	|su	|su	X
ejpam-4412	39	3	|t	|t	VERB
ejpam-4412	39	4	|t	|t	PROPN
ejpam-4412	39	5	.	.	PUNCT
ejpam-4412	40	1	moreover	moreover	ADV
ejpam-4412	40	2	,	,	PUNCT
ejpam-4412	40	3	for	for	ADP
ejpam-4412	40	4	each	each	DET
ejpam-4412	40	5	nonnegative	nonnegative	ADJ
ejpam-4412	40	6	integer	integer	NOUN
ejpam-4412	40	7	n	n	CCONJ
ejpam-4412	40	8	,	,	PUNCT
ejpam-4412	40	9	the	the	DET
ejpam-4412	40	10	n	n	ADV
ejpam-4412	40	11	-	-	PUNCT
ejpam-4412	40	12	th	th	X
ejpam-4412	40	13	generalized	generalize	VERB
ejpam-4412	40	14	aluthge	aluthge	ADJ
ejpam-4412	40	15	transform	transform	NOUN
ejpam-4412	40	16	∆n(t	∆n(t	PROPN
ejpam-4412	40	17	(	(	PUNCT
ejpam-4412	40	18	s	s	PROPN
ejpam-4412	40	19	,	,	PUNCT
ejpam-4412	40	20	t	t	PROPN
ejpam-4412	40	21	)	)	PUNCT
ejpam-4412	40	22	)	)	PUNCT
ejpam-4412	40	23	of	of	ADP
ejpam-4412	40	24	t	t	PROPN
ejpam-4412	40	25	(	(	PUNCT
ejpam-4412	40	26	s	s	PROPN
ejpam-4412	40	27	,	,	PUNCT
ejpam-4412	40	28	t	t	PROPN
ejpam-4412	40	29	)	)	PUNCT
ejpam-4412	40	30	is	be	AUX
ejpam-4412	40	31	defined	define	VERB
ejpam-4412	40	32	as	as	SCONJ
ejpam-4412	40	33	follows	follow	VERB
ejpam-4412	40	34	:	:	PUNCT
ejpam-4412	40	35	∆n(t	∆n(t	NUM
ejpam-4412	40	36	(	(	PUNCT
ejpam-4412	40	37	s	s	PROPN
ejpam-4412	40	38	,	,	PUNCT
ejpam-4412	40	39	t	t	PROPN
ejpam-4412	40	40	)	)	PUNCT
ejpam-4412	40	41	)	)	PUNCT
ejpam-4412	41	1	=	=	PRON
ejpam-4412	41	2	∆(∆n−1(t	∆(∆n−1(t	X
ejpam-4412	41	3	(	(	PUNCT
ejpam-4412	41	4	s	s	PROPN
ejpam-4412	41	5	,	,	PUNCT
ejpam-4412	41	6	t))),∆0(t	t))),∆0(t	PROPN
ejpam-4412	41	7	(	(	PUNCT
ejpam-4412	41	8	s	s	PROPN
ejpam-4412	41	9	,	,	PUNCT
ejpam-4412	41	10	t	t	PROPN
ejpam-4412	41	11	)	)	PUNCT
ejpam-4412	41	12	)	)	PUNCT
ejpam-4412	42	1	=	=	SYM
ejpam-4412	42	2	t	t	PROPN
ejpam-4412	42	3	(	(	PUNCT
ejpam-4412	42	4	s	s	PROPN
ejpam-4412	42	5	,	,	PUNCT
ejpam-4412	42	6	t	t	PROPN
ejpam-4412	42	7	)	)	PUNCT
ejpam-4412	42	8	.	.	PUNCT
ejpam-4412	43	1	definition	definition	NOUN
ejpam-4412	43	2	2	2	NUM
ejpam-4412	43	3	.	.	PUNCT
ejpam-4412	44	1	let	let	VERB
ejpam-4412	44	2	0	0	NUM
ejpam-4412	44	3	<	<	X
ejpam-4412	44	4	s	s	PROPN
ejpam-4412	44	5	,	,	PUNCT
ejpam-4412	44	6	t	t	PROPN
ejpam-4412	44	7	,	,	PUNCT
ejpam-4412	44	8	and	and	CCONJ
ejpam-4412	44	9	0	0	NUM
ejpam-4412	44	10	<	<	X
ejpam-4412	44	11	p	p	X
ejpam-4412	44	12	≤	≤	NUM
ejpam-4412	44	13	1	1	NUM
ejpam-4412	44	14	.	.	PUNCT
ejpam-4412	45	1	an	an	DET
ejpam-4412	45	2	operator	operator	NOUN
ejpam-4412	45	3	t	t	NOUN
ejpam-4412	45	4	is	be	AUX
ejpam-4412	45	5	said	say	VERB
ejpam-4412	45	6	to	to	PART
ejpam-4412	45	7	be	be	AUX
ejpam-4412	45	8	a	a	DET
ejpam-4412	45	9	class	class	NOUN
ejpam-4412	45	10	(	(	PUNCT
ejpam-4412	45	11	i	i	NOUN
ejpam-4412	45	12	)	)	PUNCT
ejpam-4412	45	13	p	p	NOUN
ejpam-4412	45	14	-	-	PUNCT
ejpam-4412	45	15	wa(s	wa(s	NUM
ejpam-4412	45	16	,	,	PUNCT
ejpam-4412	45	17	t	t	PROPN
ejpam-4412	45	18	)	)	PUNCT
ejpam-4412	45	19	if	if	SCONJ
ejpam-4412	45	20	(	(	PUNCT
ejpam-4412	45	21	|t	|t	ADJ
ejpam-4412	45	22	∗|t|t	∗|t|t	NOUN
ejpam-4412	45	23	|2s|t	|2s|t	NOUN
ejpam-4412	45	24	∗|t	∗|t	NOUN
ejpam-4412	45	25	)	)	PUNCT
ejpam-4412	45	26	tp	tp	ADP
ejpam-4412	45	27	s+t	s+t	PROPN
ejpam-4412	45	28	≥	≥	NOUN
ejpam-4412	45	29	|t	|t	PROPN
ejpam-4412	45	30	∗|2tp	∗|2tp	PUNCT
ejpam-4412	45	31	and	and	CCONJ
ejpam-4412	45	32	|t	|t	VERB
ejpam-4412	45	33	|2sp	|2sp	PROPN
ejpam-4412	45	34	≥	≥	NUM
ejpam-4412	45	35	(	(	PUNCT
ejpam-4412	45	36	|t	|t	PROPN
ejpam-4412	45	37	|s|t	|s|t	VERB
ejpam-4412	45	38	∗|2t|t	∗|2t|t	PROPN
ejpam-4412	45	39	|s	|s	NUM
ejpam-4412	45	40	)	)	PUNCT
ejpam-4412	45	41	sp	sp	ADP
ejpam-4412	45	42	s+t	s+t	PROPN
ejpam-4412	45	43	.	.	PUNCT
ejpam-4412	46	1	(	(	PUNCT
ejpam-4412	46	2	ii	ii	NOUN
ejpam-4412	46	3	)	)	PUNCT
ejpam-4412	46	4	p	p	NOUN
ejpam-4412	46	5	-	-	PUNCT
ejpam-4412	46	6	a(s	a(s	PROPN
ejpam-4412	46	7	,	,	PUNCT
ejpam-4412	46	8	t	t	PROPN
ejpam-4412	46	9	)	)	PUNCT
ejpam-4412	46	10	if	if	SCONJ
ejpam-4412	46	11	(	(	PUNCT
ejpam-4412	46	12	|t	|t	ADJ
ejpam-4412	46	13	∗|t|t	∗|t|t	NOUN
ejpam-4412	46	14	|2s|t	|2s|t	NOUN
ejpam-4412	46	15	∗|t	∗|t	NOUN
ejpam-4412	46	16	)	)	PUNCT
ejpam-4412	46	17	tp	tp	ADP
ejpam-4412	46	18	s+t	s+t	PROPN
ejpam-4412	46	19	≥	≥	NUM
ejpam-4412	46	20	|t	|t	NOUN
ejpam-4412	46	21	∗|2tp	∗|2tp	SYM
ejpam-4412	46	22	.	.	PUNCT
ejpam-4412	47	1	(	(	PUNCT
ejpam-4412	47	2	iii	iii	X
ejpam-4412	47	3	)	)	PUNCT
ejpam-4412	47	4	p	p	NOUN
ejpam-4412	47	5	-	-	PUNCT
ejpam-4412	47	6	a	a	NOUN
ejpam-4412	47	7	if	if	SCONJ
ejpam-4412	47	8	|t	|t	PROPN
ejpam-4412	47	9	2|p	2|p	NUM
ejpam-4412	47	10	≥	≥	NOUN
ejpam-4412	47	11	|t	|t	VERB
ejpam-4412	47	12	|2p	|2p	NUM
ejpam-4412	47	13	.	.	PUNCT
ejpam-4412	48	1	(	(	PUNCT
ejpam-4412	48	2	iv	iv	X
ejpam-4412	48	3	)	)	PUNCT
ejpam-4412	48	4	(	(	PUNCT
ejpam-4412	48	5	s	s	X
ejpam-4412	48	6	,	,	PUNCT
ejpam-4412	48	7	p)-w	p)-w	PROPN
ejpam-4412	48	8	-	-	PUNCT
ejpam-4412	48	9	hyponormal	hyponormal	ADJ
ejpam-4412	48	10	if	if	SCONJ
ejpam-4412	48	11	|t	|t	PROPN
ejpam-4412	48	12	(	(	PUNCT
ejpam-4412	48	13	s	s	PROPN
ejpam-4412	48	14	,	,	PUNCT
ejpam-4412	48	15	s)|p	s)|p	NOUN
ejpam-4412	48	16	≥	≥	NOUN
ejpam-4412	48	17	|t	|t	VERB
ejpam-4412	49	1	|2sp	|2sp	PROPN
ejpam-4412	49	2	≥	≥	NUM
ejpam-4412	49	3	|(t	|(t	PROPN
ejpam-4412	49	4	(	(	PUNCT
ejpam-4412	49	5	s	s	PROPN
ejpam-4412	49	6	,	,	PUNCT
ejpam-4412	49	7	s)∗|p	s)∗|p	PROPN
ejpam-4412	49	8	.	.	PUNCT
ejpam-4412	50	1	it	it	PRON
ejpam-4412	50	2	is	be	AUX
ejpam-4412	50	3	known	know	VERB
ejpam-4412	50	4	that	that	SCONJ
ejpam-4412	50	5	p	p	PROPN
ejpam-4412	50	6	-	-	PUNCT
ejpam-4412	50	7	hyponormal	hyponormal	ADJ
ejpam-4412	50	8	operators	operator	NOUN
ejpam-4412	50	9	and	and	CCONJ
ejpam-4412	50	10	log	log	NOUN
ejpam-4412	50	11	-	-	PUNCT
ejpam-4412	50	12	hyponormal	hyponormal	NOUN
ejpam-4412	50	13	operators	operator	NOUN
ejpam-4412	50	14	are	be	AUX
ejpam-4412	50	15	class	class	NOUN
ejpam-4412	50	16	1wa(s	1wa(s	NUM
ejpam-4412	50	17	,	,	PUNCT
ejpam-4412	50	18	t	t	PROPN
ejpam-4412	50	19	)	)	PUNCT
ejpam-4412	50	20	for	for	ADP
ejpam-4412	50	21	any	any	DET
ejpam-4412	50	22	0	0	PUNCT
ejpam-4412	50	23	<	<	X
ejpam-4412	50	24	s	s	PROPN
ejpam-4412	50	25	,	,	PUNCT
ejpam-4412	50	26	t.	t.	NOUN
ejpam-4412	50	27	class	class	NOUN
ejpam-4412	50	28	p	p	NOUN
ejpam-4412	50	29	-	-	PUNCT
ejpam-4412	50	30	wa(s	wa(s	NUM
ejpam-4412	50	31	,	,	PUNCT
ejpam-4412	50	32	s	s	AUX
ejpam-4412	50	33	)	)	PUNCT
ejpam-4412	50	34	is	be	AUX
ejpam-4412	50	35	called	call	VERB
ejpam-4412	50	36	class	class	NOUN
ejpam-4412	50	37	(	(	PUNCT
ejpam-4412	50	38	s	s	PROPN
ejpam-4412	50	39	,	,	PUNCT
ejpam-4412	50	40	p)-w	p)-w	PROPN
ejpam-4412	50	41	-	-	PUNCT
ejpam-4412	50	42	hyponormal	hyponormal	ADJ
ejpam-4412	50	43	,	,	PUNCT
ejpam-4412	50	44	class	class	NOUN
ejpam-4412	50	45	1wa(1	1wa(1	PROPN
ejpam-4412	50	46	,	,	PUNCT
ejpam-4412	50	47	1	1	NUM
ejpam-4412	50	48	)	)	PUNCT
ejpam-4412	50	49	is	be	AUX
ejpam-4412	50	50	called	call	VERB
ejpam-4412	50	51	class	class	NOUN
ejpam-4412	50	52	a	a	NOUN
ejpam-4412	50	53	and	and	CCONJ
ejpam-4412	50	54	class	class	NOUN
ejpam-4412	50	55	1	1	NUM
ejpam-4412	50	56	-	-	PUNCT
ejpam-4412	50	57	wa(12	wa(12	NOUN
ejpam-4412	50	58	,	,	PUNCT
ejpam-4412	50	59	1	1	NUM
ejpam-4412	50	60	2	2	NUM
ejpam-4412	50	61	)	)	PUNCT
ejpam-4412	50	62	is	be	AUX
ejpam-4412	50	63	called	call	VERB
ejpam-4412	50	64	w	w	NOUN
ejpam-4412	50	65	-	-	PUNCT
ejpam-4412	50	66	hyponormal	hyponormal	ADJ
ejpam-4412	50	67	[	[	X
ejpam-4412	50	68	13	13	NUM
ejpam-4412	50	69	,	,	PUNCT
ejpam-4412	50	70	15	15	NUM
ejpam-4412	50	71	,	,	PUNCT
ejpam-4412	50	72	18	18	NUM
ejpam-4412	50	73	,	,	PUNCT
ejpam-4412	50	74	19	19	NUM
ejpam-4412	50	75	,	,	PUNCT
ejpam-4412	50	76	33	33	NUM
ejpam-4412	50	77	]	]	PUNCT
ejpam-4412	50	78	.	.	PUNCT
ejpam-4412	51	1	hence	hence	ADV
ejpam-4412	51	2	class	class	NOUN
ejpam-4412	51	3	p	p	NOUN
ejpam-4412	51	4	-	-	PUNCT
ejpam-4412	51	5	wa(s	wa(s	NUM
ejpam-4412	51	6	,	,	PUNCT
ejpam-4412	51	7	t	t	NOUN
ejpam-4412	51	8	)	)	PUNCT
ejpam-4412	51	9	operator	operator	NOUN
ejpam-4412	51	10	is	be	AUX
ejpam-4412	51	11	a	a	DET
ejpam-4412	51	12	generalization	generalization	NOUN
ejpam-4412	51	13	of	of	ADP
ejpam-4412	51	14	class	class	NOUN
ejpam-4412	51	15	(	(	PUNCT
ejpam-4412	51	16	s	s	PROPN
ejpam-4412	51	17	,	,	PUNCT
ejpam-4412	51	18	p)-w	p)-w	PROPN
ejpam-4412	51	19	-	-	PUNCT
ejpam-4412	51	20	hyponormal	hyponormal	ADJ
ejpam-4412	51	21	,	,	PUNCT
ejpam-4412	51	22	class	class	NOUN
ejpam-4412	51	23	a	a	PRON
ejpam-4412	51	24	and	and	CCONJ
ejpam-4412	51	25	w	w	NOUN
ejpam-4412	51	26	-	-	PUNCT
ejpam-4412	51	27	hyponormal	hyponormal	ADJ
ejpam-4412	51	28	operators	operator	NOUN
ejpam-4412	51	29	.	.	PUNCT
ejpam-4412	52	1	c.	c.	PROPN
ejpam-4412	52	2	yang	yang	PROPN
ejpam-4412	52	3	and	and	CCONJ
ejpam-4412	52	4	j.	j.	PROPN
ejpam-4412	52	5	yuan	yuan	PROPN
ejpam-4412	53	1	[	[	X
ejpam-4412	53	2	34–36	34–36	NUM
ejpam-4412	53	3	]	]	PUNCT
ejpam-4412	53	4	studied	study	VERB
ejpam-4412	53	5	class	class	NOUN
ejpam-4412	53	6	wf	wf	PROPN
ejpam-4412	53	7	(	(	PUNCT
ejpam-4412	53	8	p	p	X
ejpam-4412	53	9	,	,	PUNCT
ejpam-4412	53	10	r	r	NOUN
ejpam-4412	53	11	,	,	PUNCT
ejpam-4412	53	12	q	q	NOUN
ejpam-4412	53	13	)	)	PUNCT
ejpam-4412	53	14	operator	operator	NOUN
ejpam-4412	53	15	t	t	NOUN
ejpam-4412	53	16	,	,	PUNCT
ejpam-4412	53	17	i.e.	i.e.	X
ejpam-4412	53	18	,	,	PUNCT
ejpam-4412	53	19	(	(	PUNCT
ejpam-4412	53	20	|t	|t	NOUN
ejpam-4412	53	21	∗|r|t	∗|r|t	NUM
ejpam-4412	53	22	|2p|t	|2p|t	NOUN
ejpam-4412	53	23	∗|r	∗|r	NOUN
ejpam-4412	53	24	)	)	PUNCT
ejpam-4412	53	25	1	1	NUM
ejpam-4412	53	26	q	q	NOUN
ejpam-4412	53	27	≥	≥	NOUN
ejpam-4412	53	28	|t	|t	VERB
ejpam-4412	53	29	∗|	∗|	PROPN
ejpam-4412	53	30	2(p+r	2(p+r	NUM
ejpam-4412	53	31	)	)	PUNCT
ejpam-4412	53	32	q	q	NOUN
ejpam-4412	53	33	and	and	CCONJ
ejpam-4412	53	34	|t	|t	VERB
ejpam-4412	53	35	|2(p+r)(1−	|2(p+r)(1−	NOUN
ejpam-4412	53	36	1	1	NUM
ejpam-4412	53	37	q	q	NOUN
ejpam-4412	53	38	)	)	PUNCT
ejpam-4412	53	39	≥	≥	PROPN
ejpam-4412	53	40	(	(	PUNCT
ejpam-4412	53	41	|t	|t	VERB
ejpam-4412	53	42	|p|t	|p|t	PROPN
ejpam-4412	53	43	∗|2r|t	∗|2r|t	PROPN
ejpam-4412	53	44	|p	|p	PROPN
ejpam-4412	53	45	)	)	PUNCT
ejpam-4412	53	46	1−	1−	NUM
ejpam-4412	53	47	1	1	NUM
ejpam-4412	53	48	q	q	NOUN
ejpam-4412	53	49	where	where	SCONJ
ejpam-4412	53	50	0	0	NUM
ejpam-4412	53	51	<	<	X
ejpam-4412	53	52	p	p	X
ejpam-4412	53	53	,	,	PUNCT
ejpam-4412	53	54	0	0	PUNCT
ejpam-4412	53	55	<	<	X
ejpam-4412	53	56	r	r	NOUN
ejpam-4412	53	57	,	,	PUNCT
ejpam-4412	53	58	1	1	NUM
ejpam-4412	53	59	≤	≤	NUM
ejpam-4412	53	60	q.	q.	NOUN
ejpam-4412	53	61	if	if	SCONJ
ejpam-4412	53	62	we	we	PRON
ejpam-4412	53	63	take	take	VERB
ejpam-4412	53	64	small	small	ADJ
ejpam-4412	53	65	p1	p1	NOUN
ejpam-4412	53	66	such	such	ADJ
ejpam-4412	53	67	that	that	SCONJ
ejpam-4412	53	68	0	0	PUNCT
ejpam-4412	53	69	<	<	X
ejpam-4412	53	70	p1	p1	PROPN
ejpam-4412	53	71	≤	≤	X
ejpam-4412	53	72	p+r	p+r	X
ejpam-4412	53	73	qr	qr	NOUN
ejpam-4412	53	74	and	and	CCONJ
ejpam-4412	53	75	p1	p1	PROPN
ejpam-4412	53	76	≤	≤	PROPN
ejpam-4412	53	77	(	(	PUNCT
ejpam-4412	53	78	p+r)(q−1	p+r)(q−1	NOUN
ejpam-4412	53	79	)	)	PUNCT
ejpam-4412	53	80	pq	pq	INTJ
ejpam-4412	53	81	,	,	PUNCT
ejpam-4412	53	82	then	then	ADV
ejpam-4412	53	83	t	t	PROPN
ejpam-4412	53	84	is	be	AUX
ejpam-4412	53	85	class	class	NOUN
ejpam-4412	53	86	p1	p1	NOUN
ejpam-4412	53	87	-	-	PUNCT
ejpam-4412	53	88	wa(p	wa(p	NOUN
ejpam-4412	53	89	,	,	PUNCT
ejpam-4412	53	90	r	r	NOUN
ejpam-4412	53	91	)	)	PUNCT
ejpam-4412	53	92	.	.	PUNCT
ejpam-4412	54	1	hence	hence	ADV
ejpam-4412	54	2	class	class	NOUN
ejpam-4412	54	3	p1	p1	PROPN
ejpam-4412	54	4	-	-	PUNCT
ejpam-4412	54	5	wa(p	wa(p	NOUN
ejpam-4412	54	6	,	,	PUNCT
ejpam-4412	54	7	r	r	NOUN
ejpam-4412	54	8	)	)	PUNCT
ejpam-4412	54	9	is	be	AUX
ejpam-4412	54	10	a	a	DET
ejpam-4412	54	11	generalization	generalization	NOUN
ejpam-4412	54	12	of	of	ADP
ejpam-4412	54	13	class	class	NOUN
ejpam-4412	54	14	wf	wf	PROPN
ejpam-4412	54	15	(	(	PUNCT
ejpam-4412	54	16	p	p	X
ejpam-4412	54	17	,	,	PUNCT
ejpam-4412	54	18	r	r	NOUN
ejpam-4412	54	19	,	,	PUNCT
ejpam-4412	54	20	q	q	NOUN
ejpam-4412	54	21	)	)	PUNCT
ejpam-4412	54	22	.	.	PUNCT
ejpam-4412	55	1	we	we	PRON
ejpam-4412	55	2	will	will	AUX
ejpam-4412	55	3	use	use	VERB
ejpam-4412	55	4	this	this	DET
ejpam-4412	55	5	property	property	NOUN
ejpam-4412	55	6	frequently	frequently	ADV
ejpam-4412	55	7	.	.	PUNCT
ejpam-4412	56	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	56	2	,	,	PUNCT
ejpam-4412	56	3	n.	n.	NOUN
ejpam-4412	56	4	h.	h.	PROPN
ejpam-4412	56	5	altaweel	altaweel	PROPN
ejpam-4412	56	6	/	/	SYM
ejpam-4412	56	7	eur	eur	PROPN
ejpam-4412	56	8	.	.	PUNCT
ejpam-4412	57	1	j.	j.	PROPN
ejpam-4412	57	2	pure	pure	PROPN
ejpam-4412	57	3	appl	appl	PROPN
ejpam-4412	57	4	.	.	PROPN
ejpam-4412	57	5	math	math	PROPN
ejpam-4412	57	6	,	,	PUNCT
ejpam-4412	57	7	15	15	NUM
ejpam-4412	57	8	(	(	PUNCT
ejpam-4412	57	9	3	3	NUM
ejpam-4412	57	10	)	)	PUNCT
ejpam-4412	57	11	(	(	PUNCT
ejpam-4412	57	12	2022	2022	NUM
ejpam-4412	57	13	)	)	PUNCT
ejpam-4412	57	14	,	,	PUNCT
ejpam-4412	57	15	1067	1067	NUM
ejpam-4412	57	16	-	-	SYM
ejpam-4412	57	17	1089	1089	NUM
ejpam-4412	57	18	1069	1069	NUM
ejpam-4412	57	19	it	it	PRON
ejpam-4412	57	20	is	be	AUX
ejpam-4412	57	21	known	know	VERB
ejpam-4412	57	22	that	that	SCONJ
ejpam-4412	57	23	t	t	NOUN
ejpam-4412	57	24	=	=	SYM
ejpam-4412	57	25	u	u	NOUN
ejpam-4412	57	26	|t	|t	NOUN
ejpam-4412	57	27	|	|	ADV
ejpam-4412	57	28	is	be	AUX
ejpam-4412	57	29	class	class	NOUN
ejpam-4412	57	30	p	p	NOUN
ejpam-4412	57	31	-	-	PUNCT
ejpam-4412	57	32	wa(s	wa(s	NUM
ejpam-4412	57	33	,	,	PUNCT
ejpam-4412	57	34	t	t	NOUN
ejpam-4412	57	35	)	)	PUNCT
ejpam-4412	58	1	if	if	SCONJ
ejpam-4412	58	2	and	and	CCONJ
ejpam-4412	58	3	only	only	ADV
ejpam-4412	58	4	if	if	SCONJ
ejpam-4412	58	5	|t	|t	PROPN
ejpam-4412	58	6	(	(	PUNCT
ejpam-4412	58	7	s	s	X
ejpam-4412	58	8	,	,	PUNCT
ejpam-4412	58	9	t)|	t)|	ADJ
ejpam-4412	58	10	2tp	2tp	NOUN
ejpam-4412	58	11	s+t	s+t	PROPN
ejpam-4412	58	12	≥	≥	NOUN
ejpam-4412	58	13	|t	|t	PROPN
ejpam-4412	58	14	|2tp	|2tp	PROPN
ejpam-4412	58	15	,	,	PUNCT
ejpam-4412	58	16	|t	|t	PROPN
ejpam-4412	58	17	|2sp	|2sp	PROPN
ejpam-4412	58	18	≥	≥	NUM
ejpam-4412	58	19	|t	|t	PROPN
ejpam-4412	58	20	(	(	PUNCT
ejpam-4412	58	21	s	s	PROPN
ejpam-4412	58	22	,	,	PUNCT
ejpam-4412	58	23	t)∗|	t)∗|	ADV
ejpam-4412	58	24	2sp	2sp	NOUN
ejpam-4412	58	25	s+t	s+t	INTJ
ejpam-4412	58	26	by	by	ADP
ejpam-4412	58	27	[	[	X
ejpam-4412	58	28	26	26	NUM
ejpam-4412	58	29	]	]	PUNCT
ejpam-4412	58	30	.	.	PUNCT
ejpam-4412	59	1	hence	hence	ADV
ejpam-4412	59	2	|t	|t	PROPN
ejpam-4412	59	3	(	(	PUNCT
ejpam-4412	59	4	s	s	NOUN
ejpam-4412	59	5	,	,	PUNCT
ejpam-4412	59	6	t)|	t)|	ADV
ejpam-4412	59	7	2rp	2rp	ADJ
ejpam-4412	59	8	s+t	s+t	PROPN
ejpam-4412	59	9	≥	≥	NOUN
ejpam-4412	59	10	|t	|t	VERB
ejpam-4412	60	1	|2rp	|2rp	ADJ
ejpam-4412	60	2	≥	≥	NUM
ejpam-4412	60	3	|t	|t	NOUN
ejpam-4412	60	4	(	(	PUNCT
ejpam-4412	60	5	s	s	PROPN
ejpam-4412	60	6	,	,	PUNCT
ejpam-4412	60	7	t)∗|	t)∗|	VERB
ejpam-4412	60	8	2rp	2rp	ADJ
ejpam-4412	60	9	s+t	s+t	PROPN
ejpam-4412	60	10	and	and	CCONJ
ejpam-4412	60	11	t	t	PROPN
ejpam-4412	60	12	(	(	PUNCT
ejpam-4412	60	13	s	s	PROPN
ejpam-4412	60	14	,	,	PUNCT
ejpam-4412	60	15	t	t	PROPN
ejpam-4412	60	16	)	)	PUNCT
ejpam-4412	60	17	is	be	AUX
ejpam-4412	60	18	rp	rp	NOUN
ejpam-4412	60	19	-	-	PUNCT
ejpam-4412	60	20	hyponormal	hyponormal	NOUN
ejpam-4412	60	21	for	for	ADP
ejpam-4412	60	22	all	all	DET
ejpam-4412	60	23	r	r	NOUN
ejpam-4412	60	24	∈	∈	PROPN
ejpam-4412	60	25	(	(	PUNCT
ejpam-4412	60	26	0,min{s	0,min{s	NUM
ejpam-4412	60	27	,	,	PUNCT
ejpam-4412	60	28	t	t	PROPN
ejpam-4412	60	29	}	}	PUNCT
ejpam-4412	60	30	]	]	PUNCT
ejpam-4412	60	31	.	.	PUNCT
ejpam-4412	61	1	the	the	DET
ejpam-4412	61	2	following	follow	VERB
ejpam-4412	61	3	is	be	AUX
ejpam-4412	61	4	a	a	DET
ejpam-4412	61	5	breakdown	breakdown	NOUN
ejpam-4412	61	6	of	of	ADP
ejpam-4412	61	7	the	the	DET
ejpam-4412	61	8	paper	paper	NOUN
ejpam-4412	61	9	’s	’s	PART
ejpam-4412	61	10	structure	structure	NOUN
ejpam-4412	61	11	:	:	PUNCT
ejpam-4412	61	12	in	in	ADP
ejpam-4412	61	13	section	section	NOUN
ejpam-4412	61	14	2	2	NUM
ejpam-4412	61	15	,	,	PUNCT
ejpam-4412	61	16	we	we	PRON
ejpam-4412	61	17	prove	prove	VERB
ejpam-4412	61	18	that	that	SCONJ
ejpam-4412	61	19	if	if	SCONJ
ejpam-4412	61	20	t	t	PROPN
ejpam-4412	61	21	is	be	AUX
ejpam-4412	61	22	a	a	DET
ejpam-4412	61	23	class	class	NOUN
ejpam-4412	61	24	of	of	ADP
ejpam-4412	61	25	p	p	NOUN
ejpam-4412	61	26	-	-	PUNCT
ejpam-4412	61	27	wa(s	wa(s	NUM
ejpam-4412	61	28	,	,	PUNCT
ejpam-4412	61	29	t	t	NOUN
ejpam-4412	61	30	)	)	PUNCT
ejpam-4412	61	31	operators	operator	NOUN
ejpam-4412	61	32	and	and	CCONJ
ejpam-4412	61	33	its	its	PRON
ejpam-4412	61	34	aluthge	aluthge	ADJ
ejpam-4412	61	35	transform	transform	NOUN
ejpam-4412	61	36	t	t	PROPN
ejpam-4412	61	37	(	(	PUNCT
ejpam-4412	61	38	s	s	PROPN
ejpam-4412	61	39	,	,	PUNCT
ejpam-4412	61	40	t	t	PROPN
ejpam-4412	61	41	)	)	PUNCT
ejpam-4412	61	42	is	be	AUX
ejpam-4412	61	43	quasinormal	quasinormal	ADJ
ejpam-4412	61	44	(	(	PUNCT
ejpam-4412	61	45	respectively	respectively	ADV
ejpam-4412	61	46	,	,	PUNCT
ejpam-4412	61	47	normal	normal	ADJ
ejpam-4412	61	48	)	)	PUNCT
ejpam-4412	61	49	,	,	PUNCT
ejpam-4412	61	50	then	then	ADV
ejpam-4412	61	51	t	t	PROPN
ejpam-4412	61	52	is	be	AUX
ejpam-4412	61	53	also	also	ADV
ejpam-4412	61	54	quasinormal	quasinormal	ADJ
ejpam-4412	61	55	(	(	PUNCT
ejpam-4412	61	56	resp	resp	NOUN
ejpam-4412	61	57	.	.	PUNCT
ejpam-4412	61	58	,	,	PUNCT
ejpam-4412	61	59	normal	normal	ADJ
ejpam-4412	61	60	)	)	PUNCT
ejpam-4412	61	61	.	.	PUNCT
ejpam-4412	62	1	the	the	DET
ejpam-4412	62	2	normal	normal	ADJ
ejpam-4412	62	3	parts	part	NOUN
ejpam-4412	62	4	of	of	ADP
ejpam-4412	62	5	quasisimilar	quasisimilar	ADJ
ejpam-4412	62	6	class	class	NOUN
ejpam-4412	62	7	p	p	NOUN
ejpam-4412	62	8	-	-	PUNCT
ejpam-4412	62	9	wa(s	wa(s	NUM
ejpam-4412	62	10	,	,	PUNCT
ejpam-4412	62	11	t	t	PROPN
ejpam-4412	62	12	)	)	PUNCT
ejpam-4412	62	13	operators	operator	NOUN
ejpam-4412	62	14	are	be	AUX
ejpam-4412	62	15	unitarily	unitarily	ADV
ejpam-4412	62	16	equivalent	equivalent	ADJ
ejpam-4412	62	17	in	in	ADP
ejpam-4412	62	18	section	section	NOUN
ejpam-4412	62	19	3	3	NUM
ejpam-4412	62	20	.	.	PUNCT
ejpam-4412	63	1	the	the	DET
ejpam-4412	63	2	major	major	ADJ
ejpam-4412	63	3	goal	goal	NOUN
ejpam-4412	63	4	of	of	ADP
ejpam-4412	63	5	section	section	NOUN
ejpam-4412	63	6	4	4	NUM
ejpam-4412	63	7	is	be	AUX
ejpam-4412	63	8	to	to	PART
ejpam-4412	63	9	demonstrate	demonstrate	VERB
ejpam-4412	63	10	that	that	SCONJ
ejpam-4412	63	11	the	the	DET
ejpam-4412	63	12	fuglede	fuglede	NOUN
ejpam-4412	63	13	-	-	PUNCT
ejpam-4412	63	14	putnam	putnam	NOUN
ejpam-4412	63	15	theorem	theorem	NOUN
ejpam-4412	63	16	holds	hold	VERB
ejpam-4412	63	17	for	for	ADP
ejpam-4412	63	18	a	a	DET
ejpam-4412	63	19	class	class	NOUN
ejpam-4412	63	20	p	p	NOUN
ejpam-4412	63	21	-	-	PUNCT
ejpam-4412	63	22	wa(s	wa(s	NUM
ejpam-4412	63	23	,	,	PUNCT
ejpam-4412	63	24	t	t	NOUN
ejpam-4412	63	25	)	)	PUNCT
ejpam-4412	63	26	operator	operator	NOUN
ejpam-4412	63	27	t	t	NOUN
ejpam-4412	63	28	with	with	ADP
ejpam-4412	63	29	0	0	NUM
ejpam-4412	63	30	<	<	X
ejpam-4412	63	31	s	s	PROPN
ejpam-4412	63	32	,	,	PUNCT
ejpam-4412	63	33	t	t	PROPN
ejpam-4412	63	34	,	,	PUNCT
ejpam-4412	63	35	s+t	s+t	PROPN
ejpam-4412	63	36	=	=	SYM
ejpam-4412	63	37	1	1	NUM
ejpam-4412	63	38	and	and	CCONJ
ejpam-4412	63	39	0	0	NUM
ejpam-4412	63	40	<	<	X
ejpam-4412	63	41	p	p	X
ejpam-4412	63	42	≤	≤	NOUN
ejpam-4412	63	43	1	1	NUM
ejpam-4412	63	44	if	if	SCONJ
ejpam-4412	63	45	t	t	PROPN
ejpam-4412	63	46	fulfills	fulfill	VERB
ejpam-4412	63	47	the	the	DET
ejpam-4412	63	48	kernel	kernel	PROPN
ejpam-4412	63	49	condition	condition	NOUN
ejpam-4412	63	50	ker(t	ker(t	NOUN
ejpam-4412	63	51	)	)	PUNCT
ejpam-4412	63	52	⊂	⊂	PROPN
ejpam-4412	64	1	ker(t	ker(t	NOUN
ejpam-4412	64	2	∗	∗	NOUN
ejpam-4412	64	3	)	)	PUNCT
ejpam-4412	64	4	.	.	PUNCT
ejpam-4412	65	1	2	2	X
ejpam-4412	65	2	.	.	X
ejpam-4412	65	3	quasinormality	quasinormality	NOUN
ejpam-4412	65	4	let	let	VERB
ejpam-4412	65	5	t	t	NOUN
ejpam-4412	65	6	=	=	SYM
ejpam-4412	65	7	u	u	NOUN
ejpam-4412	65	8	|t	|t	VERB
ejpam-4412	65	9	|	|	ADV
ejpam-4412	65	10	be	be	AUX
ejpam-4412	65	11	the	the	DET
ejpam-4412	65	12	polar	polar	ADJ
ejpam-4412	65	13	decomposition	decomposition	NOUN
ejpam-4412	65	14	of	of	ADP
ejpam-4412	65	15	t	t	PROPN
ejpam-4412	65	16	∈	∈	PROPN
ejpam-4412	65	17	b(h	b(h	PROPN
ejpam-4412	65	18	)	)	PUNCT
ejpam-4412	65	19	.	.	PUNCT
ejpam-4412	66	1	t	t	PROPN
ejpam-4412	66	2	is	be	AUX
ejpam-4412	66	3	said	say	VERB
ejpam-4412	66	4	to	to	PART
ejpam-4412	66	5	be	be	AUX
ejpam-4412	66	6	quasinormal	quasinormal	ADJ
ejpam-4412	66	7	if	if	SCONJ
ejpam-4412	66	8	|t	|t	PROPN
ejpam-4412	66	9	|u	|u	ADJ
ejpam-4412	66	10	=	=	SYM
ejpam-4412	66	11	u	u	NOUN
ejpam-4412	66	12	|t	|t	NOUN
ejpam-4412	66	13	|	|	ADV
ejpam-4412	66	14	,	,	PUNCT
ejpam-4412	66	15	or	or	CCONJ
ejpam-4412	66	16	equivalently	equivalently	ADV
ejpam-4412	66	17	,	,	PUNCT
ejpam-4412	66	18	tt	tt	PROPN
ejpam-4412	66	19	∗t	∗t	PROPN
ejpam-4412	66	20	=	=	SYM
ejpam-4412	66	21	t	t	PROPN
ejpam-4412	66	22	∗tt	∗tt	PROPN
ejpam-4412	66	23	.	.	PUNCT
ejpam-4412	67	1	s.	s.	PROPN
ejpam-4412	67	2	m.	m.	PROPN
ejpam-4412	67	3	patel	patel	PROPN
ejpam-4412	67	4	,	,	PUNCT
ejpam-4412	67	5	k.	k.	PROPN
ejpam-4412	67	6	tanahashi	tanahashi	PROPN
ejpam-4412	67	7	,	,	PUNCT
ejpam-4412	67	8	a.	a.	NOUN
ejpam-4412	67	9	uchiyama	uchiyama	NOUN
ejpam-4412	67	10	and	and	CCONJ
ejpam-4412	67	11	m.	m.	NOUN
ejpam-4412	67	12	yanagida	yanagida	PROPN
ejpam-4412	68	1	[	[	X
ejpam-4412	68	2	27	27	NUM
ejpam-4412	68	3	]	]	PUNCT
ejpam-4412	68	4	proved	prove	VERB
ejpam-4412	68	5	that	that	SCONJ
ejpam-4412	68	6	if	if	SCONJ
ejpam-4412	68	7	t	t	PROPN
ejpam-4412	68	8	is	be	AUX
ejpam-4412	68	9	class	class	NOUN
ejpam-4412	68	10	a(s	a(s	PROPN
ejpam-4412	68	11	,	,	PUNCT
ejpam-4412	68	12	t	t	PROPN
ejpam-4412	68	13	)	)	PUNCT
ejpam-4412	68	14	and	and	CCONJ
ejpam-4412	68	15	t	t	PROPN
ejpam-4412	68	16	(	(	PUNCT
ejpam-4412	68	17	s	s	PROPN
ejpam-4412	68	18	,	,	PUNCT
ejpam-4412	68	19	t	t	PROPN
ejpam-4412	68	20	)	)	PUNCT
ejpam-4412	68	21	is	be	AUX
ejpam-4412	68	22	quasinormal	quasinormal	ADJ
ejpam-4412	68	23	,	,	PUNCT
ejpam-4412	68	24	then	then	ADV
ejpam-4412	68	25	t	t	PROPN
ejpam-4412	68	26	is	be	AUX
ejpam-4412	68	27	quasinormal	quasinormal	ADJ
ejpam-4412	68	28	and	and	CCONJ
ejpam-4412	68	29	t	t	PROPN
ejpam-4412	68	30	=	=	SYM
ejpam-4412	68	31	t	t	PROPN
ejpam-4412	68	32	(	(	PUNCT
ejpam-4412	68	33	s	s	PROPN
ejpam-4412	68	34	,	,	PUNCT
ejpam-4412	68	35	t	t	PROPN
ejpam-4412	68	36	)	)	PUNCT
ejpam-4412	68	37	if	if	SCONJ
ejpam-4412	68	38	s+	s+	ADV
ejpam-4412	68	39	t	t	NOUN
ejpam-4412	68	40	=	=	SYM
ejpam-4412	68	41	1	1	X
ejpam-4412	68	42	.	.	PUNCT
ejpam-4412	69	1	the	the	DET
ejpam-4412	69	2	following	follow	VERB
ejpam-4412	69	3	is	be	AUX
ejpam-4412	69	4	a	a	DET
ejpam-4412	69	5	generalization	generalization	NOUN
ejpam-4412	69	6	of	of	ADP
ejpam-4412	69	7	this	this	DET
ejpam-4412	69	8	result	result	NOUN
ejpam-4412	69	9	.	.	PUNCT
ejpam-4412	70	1	theorem	theorem	NOUN
ejpam-4412	70	2	1	1	X
ejpam-4412	70	3	.	.	PUNCT
ejpam-4412	71	1	let	let	VERB
ejpam-4412	71	2	t	t	PROPN
ejpam-4412	71	3	be	be	AUX
ejpam-4412	71	4	a	a	DET
ejpam-4412	71	5	class	class	NOUN
ejpam-4412	71	6	p	p	NOUN
ejpam-4412	71	7	-	-	PUNCT
ejpam-4412	71	8	wa(s	wa(s	NUM
ejpam-4412	71	9	,	,	PUNCT
ejpam-4412	71	10	t	t	NOUN
ejpam-4412	71	11	)	)	PUNCT
ejpam-4412	71	12	operator	operator	NOUN
ejpam-4412	71	13	with	with	ADP
ejpam-4412	71	14	the	the	DET
ejpam-4412	71	15	polar	polar	ADJ
ejpam-4412	71	16	decomposition	decomposition	NOUN
ejpam-4412	71	17	t	t	NOUN
ejpam-4412	71	18	=	=	SYM
ejpam-4412	71	19	u	u	NOUN
ejpam-4412	71	20	|t	|t	NOUN
ejpam-4412	71	21	|	|	INTJ
ejpam-4412	71	22	.	.	PUNCT
ejpam-4412	72	1	if	if	SCONJ
ejpam-4412	72	2	t	t	PROPN
ejpam-4412	72	3	(	(	PUNCT
ejpam-4412	72	4	s	s	PROPN
ejpam-4412	72	5	,	,	PUNCT
ejpam-4412	72	6	t	t	PROPN
ejpam-4412	72	7	)	)	PUNCT
ejpam-4412	72	8	=	=	PUNCT
ejpam-4412	72	9	|t	|t	PROPN
ejpam-4412	72	10	|su	|su	NOUN
ejpam-4412	72	11	|t	|t	VERB
ejpam-4412	72	12	|t	|t	PROPN
ejpam-4412	72	13	is	be	AUX
ejpam-4412	72	14	quasinormal	quasinormal	ADJ
ejpam-4412	72	15	,	,	PUNCT
ejpam-4412	72	16	then	then	ADV
ejpam-4412	72	17	t	t	PROPN
ejpam-4412	72	18	is	be	AUX
ejpam-4412	72	19	also	also	ADV
ejpam-4412	72	20	quasinormal	quasinormal	ADJ
ejpam-4412	72	21	.	.	PUNCT
ejpam-4412	73	1	hence	hence	ADV
ejpam-4412	73	2	t	t	PROPN
ejpam-4412	73	3	coincides	coincide	VERB
ejpam-4412	73	4	with	with	ADP
ejpam-4412	73	5	its	its	PRON
ejpam-4412	73	6	generalized	generalized	ADJ
ejpam-4412	73	7	aluthge	aluthge	ADJ
ejpam-4412	73	8	transform	transform	NOUN
ejpam-4412	73	9	t	t	PROPN
ejpam-4412	73	10	(	(	PUNCT
ejpam-4412	73	11	s	s	PROPN
ejpam-4412	73	12	,	,	PUNCT
ejpam-4412	73	13	t	t	PROPN
ejpam-4412	73	14	)	)	PUNCT
ejpam-4412	73	15	.	.	PUNCT
ejpam-4412	74	1	proof	proof	NOUN
ejpam-4412	74	2	.	.	PUNCT
ejpam-4412	75	1	since	since	SCONJ
ejpam-4412	75	2	t	t	PROPN
ejpam-4412	75	3	is	be	AUX
ejpam-4412	75	4	a	a	DET
ejpam-4412	75	5	class	class	NOUN
ejpam-4412	75	6	p	p	NOUN
ejpam-4412	75	7	-	-	PUNCT
ejpam-4412	75	8	a(s	a(s	PROPN
ejpam-4412	75	9	,	,	PUNCT
ejpam-4412	75	10	t	t	PROPN
ejpam-4412	75	11	)	)	PUNCT
ejpam-4412	75	12	operator	operator	NOUN
ejpam-4412	75	13	,	,	PUNCT
ejpam-4412	75	14	|t	|t	PROPN
ejpam-4412	75	15	(	(	PUNCT
ejpam-4412	75	16	s	s	X
ejpam-4412	75	17	,	,	PUNCT
ejpam-4412	75	18	t)|	t)|	ADV
ejpam-4412	75	19	2rp	2rp	ADJ
ejpam-4412	75	20	s+t	s+t	PROPN
ejpam-4412	75	21	≥	≥	NOUN
ejpam-4412	75	22	|t	|t	VERB
ejpam-4412	75	23	|2rp	|2rp	ADJ
ejpam-4412	75	24	≥	≥	NUM
ejpam-4412	75	25	|(t	|(t	PROPN
ejpam-4412	75	26	(	(	PUNCT
ejpam-4412	75	27	s	s	PROPN
ejpam-4412	75	28	,	,	PUNCT
ejpam-4412	75	29	t))∗|	t))∗|	NOUN
ejpam-4412	75	30	2rp	2rp	ADJ
ejpam-4412	75	31	s+t	s+t	PROPN
ejpam-4412	75	32	(	(	PUNCT
ejpam-4412	75	33	1	1	NUM
ejpam-4412	75	34	)	)	PUNCT
ejpam-4412	75	35	for	for	ADP
ejpam-4412	75	36	all	all	DET
ejpam-4412	75	37	r	r	NOUN
ejpam-4412	75	38	∈	∈	PROPN
ejpam-4412	75	39	(	(	PUNCT
ejpam-4412	75	40	0,min{s	0,min{s	NUM
ejpam-4412	75	41	,	,	PUNCT
ejpam-4412	75	42	t	t	PROPN
ejpam-4412	75	43	}	}	PUNCT
ejpam-4412	75	44	)	)	PUNCT
ejpam-4412	75	45	by	by	ADP
ejpam-4412	75	46	[	[	X
ejpam-4412	75	47	19	19	NUM
ejpam-4412	75	48	,	,	PUNCT
ejpam-4412	75	49	theorem	theorem	VERB
ejpam-4412	75	50	3	3	NUM
ejpam-4412	75	51	]	]	PUNCT
ejpam-4412	75	52	and	and	CCONJ
ejpam-4412	75	53	löwner	löwner	NOUN
ejpam-4412	75	54	-	-	PUNCT
ejpam-4412	75	55	heinz	heinz	ADJ
ejpam-4412	75	56	inequality	inequality	NOUN
ejpam-4412	75	57	.	.	PUNCT
ejpam-4412	76	1	then	then	ADV
ejpam-4412	76	2	douglas	douglas	PROPN
ejpam-4412	76	3	’s	’s	PART
ejpam-4412	76	4	theorem	theorem	PROPN
ejpam-4412	76	5	[	[	X
ejpam-4412	76	6	11	11	NUM
ejpam-4412	76	7	]	]	PUNCT
ejpam-4412	76	8	implies	imply	VERB
ejpam-4412	76	9	ran(t	ran(t	PROPN
ejpam-4412	76	10	(	(	PUNCT
ejpam-4412	76	11	s	s	PROPN
ejpam-4412	76	12	,	,	PUNCT
ejpam-4412	76	13	t	t	PROPN
ejpam-4412	76	14	)	)	PUNCT
ejpam-4412	76	15	)	)	PUNCT
ejpam-4412	77	1	=	=	PRON
ejpam-4412	77	2	ran((|t	ran((|t	NOUN
ejpam-4412	77	3	(	(	PUNCT
ejpam-4412	77	4	s	s	NOUN
ejpam-4412	77	5	,	,	PUNCT
ejpam-4412	77	6	t))∗|	t))∗|	NOUN
ejpam-4412	77	7	)	)	PUNCT
ejpam-4412	78	1	⊂	⊂	PRON
ejpam-4412	78	2	ran(|t	ran(|t	ADV
ejpam-4412	78	3	|	|	ADV
ejpam-4412	78	4	)	)	PUNCT
ejpam-4412	79	1	=	=	PUNCT
ejpam-4412	80	1	ran(|t	ran(|t	ADV
ejpam-4412	80	2	(	(	PUNCT
ejpam-4412	80	3	s	s	X
ejpam-4412	80	4	,	,	PUNCT
ejpam-4412	80	5	t)|	t)|	NOUN
ejpam-4412	80	6	)	)	PUNCT
ejpam-4412	80	7	where	where	SCONJ
ejpam-4412	80	8	m	m	PROPN
ejpam-4412	80	9	denotes	denote	VERB
ejpam-4412	80	10	the	the	DET
ejpam-4412	80	11	norm	norm	NOUN
ejpam-4412	80	12	closure	closure	NOUN
ejpam-4412	80	13	of	of	ADP
ejpam-4412	80	14	m	m	PROPN
ejpam-4412	80	15	.	.	PUNCT
ejpam-4412	81	1	let	let	VERB
ejpam-4412	81	2	t	t	PROPN
ejpam-4412	81	3	(	(	PUNCT
ejpam-4412	81	4	s	s	PROPN
ejpam-4412	81	5	,	,	PUNCT
ejpam-4412	81	6	t	t	PROPN
ejpam-4412	81	7	)	)	PUNCT
ejpam-4412	81	8	=	=	SYM
ejpam-4412	82	1	w	w	X
ejpam-4412	82	2	|t	|t	PROPN
ejpam-4412	82	3	(	(	PUNCT
ejpam-4412	82	4	s	s	X
ejpam-4412	82	5	,	,	PUNCT
ejpam-4412	82	6	t)|	t)|	INTJ
ejpam-4412	82	7	be	be	AUX
ejpam-4412	82	8	the	the	DET
ejpam-4412	82	9	polar	polar	ADJ
ejpam-4412	82	10	decomposition	decomposition	NOUN
ejpam-4412	82	11	of	of	ADP
ejpam-4412	82	12	t	t	PROPN
ejpam-4412	82	13	(	(	PUNCT
ejpam-4412	82	14	s	s	PROPN
ejpam-4412	82	15	,	,	PUNCT
ejpam-4412	82	16	t	t	PROPN
ejpam-4412	82	17	)	)	PUNCT
ejpam-4412	82	18	.	.	PUNCT
ejpam-4412	83	1	then	then	ADV
ejpam-4412	83	2	e	e	X
ejpam-4412	83	3	:	:	PUNCT
ejpam-4412	83	4	=	=	SYM
ejpam-4412	83	5	w	w	PROPN
ejpam-4412	83	6	∗w	∗w	PROPN
ejpam-4412	83	7	=	=	SYM
ejpam-4412	83	8	u∗u	u∗u	NUM
ejpam-4412	83	9	≥	≥	NOUN
ejpam-4412	83	10	ww	ww	PROPN
ejpam-4412	83	11	∗	∗	NOUN
ejpam-4412	83	12	=	=	PROPN
ejpam-4412	83	13	:	:	PUNCT
ejpam-4412	83	14	f	f	X
ejpam-4412	83	15	.	.	PUNCT
ejpam-4412	84	1	put	put	VERB
ejpam-4412	84	2	|(t	|(t	PROPN
ejpam-4412	84	3	(	(	PUNCT
ejpam-4412	84	4	s	s	PROPN
ejpam-4412	84	5	,	,	PUNCT
ejpam-4412	84	6	t))∗|	t))∗|	NOUN
ejpam-4412	84	7	1	1	NUM
ejpam-4412	84	8	s+t	s+t	NOUN
ejpam-4412	84	9	=	=	PUNCT
ejpam-4412	84	10	(	(	PUNCT
ejpam-4412	84	11	x	x	SYM
ejpam-4412	84	12	0	0	NUM
ejpam-4412	84	13	0	0	NUM
ejpam-4412	84	14	0	0	NUM
ejpam-4412	84	15	)	)	PUNCT
ejpam-4412	84	16	,	,	PUNCT
ejpam-4412	84	17	w	w	NOUN
ejpam-4412	84	18	=	=	SYM
ejpam-4412	84	19	(	(	PUNCT
ejpam-4412	84	20	w1	w1	NOUN
ejpam-4412	84	21	w2	w2	NOUN
ejpam-4412	84	22	0	0	NUM
ejpam-4412	84	23	0	0	NUM
ejpam-4412	84	24	)	)	PUNCT
ejpam-4412	84	25	on	on	ADP
ejpam-4412	84	26	h	h	NOUN
ejpam-4412	84	27	=	=	SYM
ejpam-4412	84	28	ran(t	ran(t	X
ejpam-4412	84	29	(	(	PUNCT
ejpam-4412	84	30	s	s	PROPN
ejpam-4412	84	31	,	,	PUNCT
ejpam-4412	84	32	t))⊕	t))⊕	ADJ
ejpam-4412	84	33	ker((t	ker((t	X
ejpam-4412	84	34	(	(	PUNCT
ejpam-4412	84	35	s	s	PROPN
ejpam-4412	84	36	,	,	PUNCT
ejpam-4412	84	37	t))∗	t))∗	PROPN
ejpam-4412	84	38	)	)	PUNCT
ejpam-4412	84	39	.	.	PUNCT
ejpam-4412	85	1	then	then	ADV
ejpam-4412	85	2	x	x	PRON
ejpam-4412	85	3	is	be	AUX
ejpam-4412	85	4	injective	injective	ADJ
ejpam-4412	85	5	and	and	CCONJ
ejpam-4412	85	6	has	have	VERB
ejpam-4412	85	7	a	a	DET
ejpam-4412	85	8	dense	dense	ADJ
ejpam-4412	85	9	range	range	NOUN
ejpam-4412	85	10	.	.	PUNCT
ejpam-4412	86	1	since	since	SCONJ
ejpam-4412	86	2	t	t	PROPN
ejpam-4412	86	3	(	(	PUNCT
ejpam-4412	86	4	s	s	PROPN
ejpam-4412	86	5	,	,	PUNCT
ejpam-4412	86	6	t	t	PROPN
ejpam-4412	86	7	)	)	PUNCT
ejpam-4412	86	8	is	be	AUX
ejpam-4412	86	9	quasinormal	quasinormal	ADJ
ejpam-4412	86	10	,	,	PUNCT
ejpam-4412	86	11	w	w	ADJ
ejpam-4412	86	12	commutes	commute	NOUN
ejpam-4412	86	13	with	with	ADP
ejpam-4412	86	14	|t	|t	PROPN
ejpam-4412	86	15	(	(	PUNCT
ejpam-4412	86	16	s	s	NOUN
ejpam-4412	86	17	,	,	PUNCT
ejpam-4412	86	18	t)|	t)|	NOUN
ejpam-4412	86	19	and	and	CCONJ
ejpam-4412	86	20	|t	|t	PROPN
ejpam-4412	86	21	(	(	PUNCT
ejpam-4412	86	22	s	s	NOUN
ejpam-4412	86	23	,	,	PUNCT
ejpam-4412	86	24	t)|	t)|	ADV
ejpam-4412	86	25	2rp	2rp	ADJ
ejpam-4412	86	26	s+t	s+t	PROPN
ejpam-4412	87	1	=	=	SYM
ejpam-4412	87	2	w	w	PROPN
ejpam-4412	87	3	∗w	∗w	PROPN
ejpam-4412	87	4	|t	|t	VERB
ejpam-4412	87	5	(	(	PUNCT
ejpam-4412	87	6	s	s	NOUN
ejpam-4412	87	7	,	,	PUNCT
ejpam-4412	87	8	t)|	t)|	ADV
ejpam-4412	87	9	2rp	2rp	ADJ
ejpam-4412	87	10	s+t	s+t	PROPN
ejpam-4412	87	11	=	=	SYM
ejpam-4412	87	12	w	w	PROPN
ejpam-4412	87	13	∗|t	∗|t	PROPN
ejpam-4412	87	14	(	(	PUNCT
ejpam-4412	87	15	s	s	NOUN
ejpam-4412	87	16	,	,	PUNCT
ejpam-4412	87	17	t)|	t)|	NOUN
ejpam-4412	87	18	2rp	2rp	ADJ
ejpam-4412	87	19	s+tw	s+tw	NOUN
ejpam-4412	88	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	88	2	,	,	PUNCT
ejpam-4412	88	3	n.	n.	NOUN
ejpam-4412	88	4	h.	h.	PROPN
ejpam-4412	88	5	altaweel	altaweel	PROPN
ejpam-4412	88	6	/	/	SYM
ejpam-4412	88	7	eur	eur	PROPN
ejpam-4412	88	8	.	.	PUNCT
ejpam-4412	89	1	j.	j.	PROPN
ejpam-4412	89	2	pure	pure	PROPN
ejpam-4412	89	3	appl	appl	PROPN
ejpam-4412	89	4	.	.	PROPN
ejpam-4412	89	5	math	math	PROPN
ejpam-4412	89	6	,	,	PUNCT
ejpam-4412	89	7	15	15	NUM
ejpam-4412	89	8	(	(	PUNCT
ejpam-4412	89	9	3	3	NUM
ejpam-4412	89	10	)	)	PUNCT
ejpam-4412	89	11	(	(	PUNCT
ejpam-4412	89	12	2022	2022	NUM
ejpam-4412	89	13	)	)	PUNCT
ejpam-4412	89	14	,	,	PUNCT
ejpam-4412	89	15	1067	1067	NUM
ejpam-4412	89	16	-	-	SYM
ejpam-4412	89	17	1089	1089	NUM
ejpam-4412	89	18	1070	1070	NUM
ejpam-4412	89	19	≥	≥	NOUN
ejpam-4412	89	20	w	w	PROPN
ejpam-4412	89	21	∗|t	∗|t	PROPN
ejpam-4412	89	22	|2rpw	|2rpw	PROPN
ejpam-4412	89	23	≥	≥	PROPN
ejpam-4412	89	24	w	w	PROPN
ejpam-4412	89	25	∗|(t	∗|(t	PROPN
ejpam-4412	89	26	(	(	PUNCT
ejpam-4412	89	27	s	s	X
ejpam-4412	89	28	,	,	PUNCT
ejpam-4412	90	1	t))∗|	t))∗|	NOUN
ejpam-4412	90	2	2rp	2rp	ADJ
ejpam-4412	90	3	s+tw	s+tw	PROPN
ejpam-4412	91	1	=	=	PRON
ejpam-4412	91	2	|t	|t	PROPN
ejpam-4412	91	3	(	(	PUNCT
ejpam-4412	91	4	s	s	NOUN
ejpam-4412	91	5	,	,	PUNCT
ejpam-4412	91	6	t)|	t)|	ADV
ejpam-4412	91	7	2rp	2rp	ADJ
ejpam-4412	91	8	s+t	s+t	PROPN
ejpam-4412	91	9	.	.	PUNCT
ejpam-4412	92	1	hence	hence	ADV
ejpam-4412	92	2	|t	|t	VERB
ejpam-4412	92	3	(	(	PUNCT
ejpam-4412	92	4	s	s	NOUN
ejpam-4412	92	5	,	,	PUNCT
ejpam-4412	92	6	t)|	t)|	ADV
ejpam-4412	92	7	2rp	2rp	ADJ
ejpam-4412	92	8	s+t	s+t	PROPN
ejpam-4412	93	1	=	=	SYM
ejpam-4412	93	2	w	w	PROPN
ejpam-4412	93	3	∗|t	∗|t	PROPN
ejpam-4412	93	4	(	(	PUNCT
ejpam-4412	93	5	s	s	NOUN
ejpam-4412	93	6	,	,	PUNCT
ejpam-4412	93	7	t)|	t)|	NOUN
ejpam-4412	93	8	2rp	2rp	ADJ
ejpam-4412	93	9	s+tw	s+tw	NOUN
ejpam-4412	94	1	=	=	PUNCT
ejpam-4412	94	2	w	w	PROPN
ejpam-4412	94	3	∗|t	∗|t	PROPN
ejpam-4412	94	4	|2rpw	|2rpw	NUM
ejpam-4412	94	5	,	,	PUNCT
ejpam-4412	94	6	and	and	CCONJ
ejpam-4412	94	7	|(t	|(t	PROPN
ejpam-4412	94	8	(	(	PUNCT
ejpam-4412	94	9	s	s	PROPN
ejpam-4412	94	10	,	,	PUNCT
ejpam-4412	94	11	t))∗|	t))∗|	NOUN
ejpam-4412	94	12	2rp	2rp	ADJ
ejpam-4412	94	13	s+t	s+t	PROPN
ejpam-4412	95	1	=	=	SYM
ejpam-4412	95	2	w	w	PROPN
ejpam-4412	95	3	|t	|t	PROPN
ejpam-4412	95	4	(	(	PUNCT
ejpam-4412	95	5	s	s	NOUN
ejpam-4412	95	6	,	,	PUNCT
ejpam-4412	95	7	t)|	t)|	ADJ
ejpam-4412	95	8	2rp	2rp	ADJ
ejpam-4412	95	9	s+tw	s+tw	NOUN
ejpam-4412	95	10	∗	∗	NOUN
ejpam-4412	95	11	=	=	SYM
ejpam-4412	95	12	ww	ww	PROPN
ejpam-4412	95	13	∗|t	∗|t	PROPN
ejpam-4412	95	14	(	(	PUNCT
ejpam-4412	95	15	s	s	X
ejpam-4412	95	16	,	,	PUNCT
ejpam-4412	95	17	t)|	t)|	NOUN
ejpam-4412	95	18	2rp	2rp	ADJ
ejpam-4412	95	19	s+tww	s+tww	NOUN
ejpam-4412	95	20	∗	∗	NOUN
ejpam-4412	95	21	(	(	PUNCT
ejpam-4412	95	22	2	2	NUM
ejpam-4412	95	23	)	)	PUNCT
ejpam-4412	95	24	=	=	SYM
ejpam-4412	95	25	ww	ww	PROPN
ejpam-4412	95	26	∗|t	∗|t	NOUN
ejpam-4412	95	27	|2rpww	|2rpww	PROPN
ejpam-4412	95	28	∗	∗	NOUN
ejpam-4412	95	29	=	=	PUNCT
ejpam-4412	95	30	(	(	PUNCT
ejpam-4412	95	31	x2rp	x2rp	X
ejpam-4412	95	32	0	0	NUM
ejpam-4412	95	33	0	0	NUM
ejpam-4412	95	34	0	0	NUM
ejpam-4412	95	35	)	)	PUNCT
ejpam-4412	95	36	.	.	PUNCT
ejpam-4412	96	1	(	(	PUNCT
ejpam-4412	96	2	3	3	X
ejpam-4412	96	3	)	)	PUNCT
ejpam-4412	96	4	since	since	SCONJ
ejpam-4412	96	5	ww	ww	PROPN
ejpam-4412	96	6	∗	∗	NOUN
ejpam-4412	96	7	=	=	PUNCT
ejpam-4412	96	8	(	(	PUNCT
ejpam-4412	96	9	1	1	NUM
ejpam-4412	96	10	0	0	NUM
ejpam-4412	96	11	0	0	NUM
ejpam-4412	96	12	0	0	NUM
ejpam-4412	96	13	)	)	PUNCT
ejpam-4412	96	14	,	,	PUNCT
ejpam-4412	96	15	(	(	PUNCT
ejpam-4412	96	16	1	1	NUM
ejpam-4412	96	17	)	)	PUNCT
ejpam-4412	96	18	,	,	PUNCT
ejpam-4412	96	19	(	(	PUNCT
ejpam-4412	96	20	2	2	X
ejpam-4412	96	21	)	)	PUNCT
ejpam-4412	96	22	and	and	CCONJ
ejpam-4412	96	23	(	(	PUNCT
ejpam-4412	96	24	3	3	X
ejpam-4412	96	25	)	)	PUNCT
ejpam-4412	96	26	imply	imply	VERB
ejpam-4412	96	27	that	that	SCONJ
ejpam-4412	96	28	|t	|t	PROPN
ejpam-4412	97	1	(	(	PUNCT
ejpam-4412	97	2	s	s	NOUN
ejpam-4412	97	3	,	,	PUNCT
ejpam-4412	97	4	t)|	t)|	ADV
ejpam-4412	97	5	2rp	2rp	ADJ
ejpam-4412	97	6	s+t	s+t	PROPN
ejpam-4412	97	7	and	and	CCONJ
ejpam-4412	97	8	|t	|t	VERB
ejpam-4412	97	9	|2rp	|2rp	PROPN
ejpam-4412	97	10	are	be	AUX
ejpam-4412	97	11	of	of	ADP
ejpam-4412	97	12	the	the	DET
ejpam-4412	97	13	forms	form	NOUN
ejpam-4412	97	14	|t	|t	VERB
ejpam-4412	97	15	(	(	PUNCT
ejpam-4412	97	16	s	s	NOUN
ejpam-4412	97	17	,	,	PUNCT
ejpam-4412	97	18	t)|	t)|	ADV
ejpam-4412	97	19	2rp	2rp	ADJ
ejpam-4412	97	20	s+t	s+t	PROPN
ejpam-4412	98	1	=	=	PUNCT
ejpam-4412	98	2	(	(	PUNCT
ejpam-4412	98	3	x2rp	x2rp	PROPN
ejpam-4412	98	4	0	0	NUM
ejpam-4412	98	5	0	0	NUM
ejpam-4412	98	6	y	y	PROPN
ejpam-4412	98	7	2rp	2rp	NOUN
ejpam-4412	98	8	)	)	PUNCT
ejpam-4412	98	9	≥	≥	NOUN
ejpam-4412	98	10	|t	|t	VERB
ejpam-4412	99	1	|2rp	|2rp	PROPN
ejpam-4412	99	2	=	=	SYM
ejpam-4412	99	3	(	(	PUNCT
ejpam-4412	99	4	x2rp	x2rp	X
ejpam-4412	99	5	0	0	NUM
ejpam-4412	99	6	0	0	NUM
ejpam-4412	99	7	z2rp	z2rp	NUM
ejpam-4412	99	8	)	)	PUNCT
ejpam-4412	99	9	,	,	PUNCT
ejpam-4412	99	10	(	(	PUNCT
ejpam-4412	99	11	4	4	X
ejpam-4412	99	12	)	)	PUNCT
ejpam-4412	99	13	where	where	SCONJ
ejpam-4412	99	14	ran(y	ran(y	X
ejpam-4412	99	15	)	)	PUNCT
ejpam-4412	100	1	=	=	PUNCT
ejpam-4412	100	2	ran(z	ran(z	PROPN
ejpam-4412	100	3	)	)	PUNCT
ejpam-4412	100	4	=	=	SYM
ejpam-4412	101	1	ran(|t	ran(|t	PROPN
ejpam-4412	101	2	|)⊖	|)⊖	NOUN
ejpam-4412	101	3	ran(t	ran(t	X
ejpam-4412	101	4	(	(	PUNCT
ejpam-4412	101	5	s	s	PROPN
ejpam-4412	101	6	,	,	PUNCT
ejpam-4412	101	7	t	t	PROPN
ejpam-4412	101	8	)	)	PUNCT
ejpam-4412	101	9	)	)	PUNCT
ejpam-4412	102	1	=	=	PUNCT
ejpam-4412	102	2	ker((t	ker((t	X
ejpam-4412	102	3	(	(	PUNCT
ejpam-4412	102	4	s	s	X
ejpam-4412	102	5	,	,	PUNCT
ejpam-4412	102	6	t))∗)⊖	t))∗)⊖	PROPN
ejpam-4412	102	7	ker(t	ker(t	PROPN
ejpam-4412	102	8	)	)	PUNCT
ejpam-4412	102	9	.	.	PUNCT
ejpam-4412	103	1	since	since	SCONJ
ejpam-4412	103	2	w	w	NOUN
ejpam-4412	103	3	commutes	commute	NOUN
ejpam-4412	103	4	with	with	ADP
ejpam-4412	103	5	|t	|t	PROPN
ejpam-4412	103	6	(	(	PUNCT
ejpam-4412	103	7	s	s	PROPN
ejpam-4412	103	8	,	,	PUNCT
ejpam-4412	103	9	t)|	t)|	INTJ
ejpam-4412	103	10	,	,	PUNCT
ejpam-4412	103	11	(	(	PUNCT
ejpam-4412	103	12	w1	w1	NOUN
ejpam-4412	103	13	w2	w2	NOUN
ejpam-4412	103	14	0	0	NUM
ejpam-4412	103	15	0	0	NUM
ejpam-4412	103	16	)	)	PUNCT
ejpam-4412	103	17	(	(	PUNCT
ejpam-4412	103	18	x	x	SYM
ejpam-4412	103	19	0	0	NUM
ejpam-4412	103	20	0	0	NUM
ejpam-4412	103	21	y	y	NOUN
ejpam-4412	103	22	)	)	PUNCT
ejpam-4412	103	23	=	=	PUNCT
ejpam-4412	104	1	(	(	PUNCT
ejpam-4412	104	2	x	x	SYM
ejpam-4412	104	3	0	0	NUM
ejpam-4412	104	4	0	0	NUM
ejpam-4412	104	5	y	y	PROPN
ejpam-4412	104	6	)	)	PUNCT
ejpam-4412	104	7	(	(	PUNCT
ejpam-4412	104	8	w1	w1	NOUN
ejpam-4412	104	9	w2	w2	NOUN
ejpam-4412	104	10	0	0	NUM
ejpam-4412	104	11	0	0	NUM
ejpam-4412	104	12	)	)	PUNCT
ejpam-4412	104	13	.	.	PUNCT
ejpam-4412	105	1	so	so	ADV
ejpam-4412	105	2	w1x	w1x	PROPN
ejpam-4412	105	3	=	=	SYM
ejpam-4412	105	4	xw1	xw1	PROPN
ejpam-4412	105	5	and	and	CCONJ
ejpam-4412	105	6	w2y	w2y	PROPN
ejpam-4412	105	7	=	=	SYM
ejpam-4412	105	8	xw2	xw2	NOUN
ejpam-4412	105	9	,	,	PUNCT
ejpam-4412	105	10	and	and	CCONJ
ejpam-4412	105	11	hence	hence	ADV
ejpam-4412	105	12	ran(w1	ran(w1	PROPN
ejpam-4412	105	13	)	)	PUNCT
ejpam-4412	105	14	and	and	CCONJ
ejpam-4412	105	15	ran(w2	ran(w2	NOUN
ejpam-4412	105	16	)	)	PUNCT
ejpam-4412	105	17	are	be	AUX
ejpam-4412	105	18	reducing	reduce	VERB
ejpam-4412	105	19	subspaces	subspace	NOUN
ejpam-4412	105	20	of	of	ADP
ejpam-4412	105	21	x	x	X
ejpam-4412	105	22	.	.	PUNCT
ejpam-4412	106	1	since	since	SCONJ
ejpam-4412	106	2	w	w	PROPN
ejpam-4412	106	3	∗w	∗w	PROPN
ejpam-4412	106	4	|t	|t	PROPN
ejpam-4412	106	5	(	(	PUNCT
ejpam-4412	106	6	s	s	X
ejpam-4412	106	7	,	,	PUNCT
ejpam-4412	106	8	t)|	t)|	NOUN
ejpam-4412	106	9	=	=	PUNCT
ejpam-4412	106	10	|t	|t	PROPN
ejpam-4412	106	11	(	(	PUNCT
ejpam-4412	106	12	s	s	PROPN
ejpam-4412	106	13	,	,	PUNCT
ejpam-4412	106	14	t)|	t)|	INTJ
ejpam-4412	106	15	,	,	PUNCT
ejpam-4412	106	16	we	we	PRON
ejpam-4412	106	17	have	have	VERB
ejpam-4412	106	18	w	w	NOUN
ejpam-4412	106	19	∗	∗	NOUN
ejpam-4412	106	20	1w1	1w1	NUM
ejpam-4412	106	21	=	=	SYM
ejpam-4412	106	22	1	1	NUM
ejpam-4412	106	23	and	and	CCONJ
ejpam-4412	106	24	xk	xk	NOUN
ejpam-4412	107	1	=	=	PUNCT
ejpam-4412	107	2	w	w	PROPN
ejpam-4412	107	3	∗	∗	NOUN
ejpam-4412	107	4	1w1x	1w1x	NOUN
ejpam-4412	108	1	k	k	NOUN
ejpam-4412	108	2	=	=	PUNCT
ejpam-4412	108	3	w	w	PROPN
ejpam-4412	108	4	∗	∗	X
ejpam-4412	108	5	1x	1x	PROPN
ejpam-4412	108	6	kw1	kw1	PROPN
ejpam-4412	108	7	,	,	PUNCT
ejpam-4412	108	8	y	y	PROPN
ejpam-4412	108	9	k	k	PROPN
ejpam-4412	109	1	=	=	PUNCT
ejpam-4412	109	2	w	w	PROPN
ejpam-4412	109	3	∗	∗	NOUN
ejpam-4412	109	4	2w2y	2w2y	NOUN
ejpam-4412	110	1	k	k	NOUN
ejpam-4412	110	2	=	=	PUNCT
ejpam-4412	110	3	w	w	PROPN
ejpam-4412	110	4	∗	∗	NOUN
ejpam-4412	110	5	2x	2x	NUM
ejpam-4412	110	6	kw2	kw2	NOUN
ejpam-4412	110	7	,	,	PUNCT
ejpam-4412	110	8	for	for	ADP
ejpam-4412	110	9	k	k	PROPN
ejpam-4412	110	10	=	=	SYM
ejpam-4412	110	11	1	1	NUM
ejpam-4412	110	12	,	,	PUNCT
ejpam-4412	110	13	2	2	NUM
ejpam-4412	110	14	,	,	PUNCT
ejpam-4412	110	15	·	·	PUNCT
ejpam-4412	110	16	·	·	PUNCT
ejpam-4412	110	17	·	·	PUNCT
ejpam-4412	110	18	.	.	PUNCT
ejpam-4412	111	1	put	put	VERB
ejpam-4412	111	2	u	u	NOUN
ejpam-4412	111	3	=	=	PUNCT
ejpam-4412	111	4	(	(	PUNCT
ejpam-4412	111	5	u11	u11	PROPN
ejpam-4412	111	6	u12	u12	PROPN
ejpam-4412	111	7	u21	u21	PROPN
ejpam-4412	111	8	u22	u22	PROPN
ejpam-4412	111	9	)	)	PUNCT
ejpam-4412	111	10	.	.	PUNCT
ejpam-4412	112	1	then	then	ADV
ejpam-4412	112	2	t	t	PROPN
ejpam-4412	112	3	(	(	PUNCT
ejpam-4412	112	4	s	s	PROPN
ejpam-4412	112	5	,	,	PUNCT
ejpam-4412	112	6	t	t	PROPN
ejpam-4412	112	7	)	)	PUNCT
ejpam-4412	112	8	=	=	PUNCT
ejpam-4412	112	9	|t	|t	PROPN
ejpam-4412	112	10	|su	|su	NOUN
ejpam-4412	112	11	|t	|t	VERB
ejpam-4412	112	12	|t	|t	PROPN
ejpam-4412	113	1	=	=	SYM
ejpam-4412	114	1	w	w	X
ejpam-4412	114	2	|t	|t	PROPN
ejpam-4412	114	3	(	(	PUNCT
ejpam-4412	114	4	s	s	AUX
ejpam-4412	114	5	,	,	PUNCT
ejpam-4412	114	6	t)|	t)|	NOUN
ejpam-4412	114	7	implies	imply	VERB
ejpam-4412	114	8	(	(	PUNCT
ejpam-4412	114	9	xs	xs	PROPN
ejpam-4412	114	10	0	0	NUM
ejpam-4412	114	11	0	0	NUM
ejpam-4412	114	12	zs	zs	PROPN
ejpam-4412	114	13	)	)	PUNCT
ejpam-4412	114	14	(	(	PUNCT
ejpam-4412	114	15	u11	u11	PROPN
ejpam-4412	114	16	u12	u12	PROPN
ejpam-4412	114	17	u21	u21	PROPN
ejpam-4412	114	18	u22	u22	PROPN
ejpam-4412	114	19	)	)	PUNCT
ejpam-4412	114	20	(	(	PUNCT
ejpam-4412	114	21	xt	xt	ADP
ejpam-4412	114	22	0	0	NUM
ejpam-4412	114	23	0	0	NUM
ejpam-4412	114	24	zt	zt	PROPN
ejpam-4412	114	25	)	)	PUNCT
ejpam-4412	114	26	=	=	PRON
ejpam-4412	115	1	(	(	PUNCT
ejpam-4412	115	2	w1	w1	NOUN
ejpam-4412	115	3	w2	w2	NOUN
ejpam-4412	115	4	0	0	NUM
ejpam-4412	115	5	0	0	NUM
ejpam-4412	115	6	)	)	PUNCT
ejpam-4412	116	1	(	(	PUNCT
ejpam-4412	116	2	xs+t	xs+t	SYM
ejpam-4412	116	3	0	0	NUM
ejpam-4412	116	4	0	0	NUM
ejpam-4412	116	5	y	y	PROPN
ejpam-4412	116	6	s+t	s+t	PROPN
ejpam-4412	116	7	)	)	PUNCT
ejpam-4412	116	8	.	.	PUNCT
ejpam-4412	117	1	hence	hence	ADV
ejpam-4412	117	2	xsu11x	xsu11x	X
ejpam-4412	117	3	t	t	NOUN
ejpam-4412	117	4	=	=	SYM
ejpam-4412	117	5	w1x	w1x	PROPN
ejpam-4412	117	6	s+t	s+t	NUM
ejpam-4412	117	7	=	=	PUNCT
ejpam-4412	117	8	xsw1x	xsw1x	PROPN
ejpam-4412	117	9	t	t	PROPN
ejpam-4412	117	10	,	,	PUNCT
ejpam-4412	117	11	xsu12z	xsu12z	PROPN
ejpam-4412	117	12	t	t	PROPN
ejpam-4412	117	13	=	=	PUNCT
ejpam-4412	117	14	w2y	w2y	PROPN
ejpam-4412	117	15	s+t	s+t	PROPN
ejpam-4412	117	16	=	=	SYM
ejpam-4412	117	17	xs+tw2	xs+tw2	PROPN
ejpam-4412	117	18	and	and	CCONJ
ejpam-4412	117	19	xs(u11	xs(u11	PUNCT
ejpam-4412	117	20	−w1)x	−w1)x	NOUN
ejpam-4412	117	21	t	t	NOUN
ejpam-4412	117	22	=	=	SYM
ejpam-4412	117	23	0	0	NUM
ejpam-4412	117	24	,	,	PUNCT
ejpam-4412	117	25	xs(u12z	xs(u12z	NUM
ejpam-4412	117	26	t	t	PROPN
ejpam-4412	117	27	−xtw2	−xtw2	PUNCT
ejpam-4412	117	28	)	)	PUNCT
ejpam-4412	118	1	=	=	PUNCT
ejpam-4412	118	2	0	0	X
ejpam-4412	118	3	.	.	PUNCT
ejpam-4412	119	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	119	2	,	,	PUNCT
ejpam-4412	119	3	n.	n.	NOUN
ejpam-4412	119	4	h.	h.	PROPN
ejpam-4412	119	5	altaweel	altaweel	PROPN
ejpam-4412	119	6	/	/	SYM
ejpam-4412	119	7	eur	eur	PROPN
ejpam-4412	119	8	.	.	PUNCT
ejpam-4412	120	1	j.	j.	PROPN
ejpam-4412	120	2	pure	pure	PROPN
ejpam-4412	120	3	appl	appl	PROPN
ejpam-4412	120	4	.	.	PROPN
ejpam-4412	120	5	math	math	PROPN
ejpam-4412	120	6	,	,	PUNCT
ejpam-4412	120	7	15	15	NUM
ejpam-4412	120	8	(	(	PUNCT
ejpam-4412	120	9	3	3	NUM
ejpam-4412	120	10	)	)	PUNCT
ejpam-4412	120	11	(	(	PUNCT
ejpam-4412	120	12	2022	2022	NUM
ejpam-4412	120	13	)	)	PUNCT
ejpam-4412	120	14	,	,	PUNCT
ejpam-4412	120	15	1067	1067	NUM
ejpam-4412	120	16	-	-	SYM
ejpam-4412	120	17	1089	1089	NUM
ejpam-4412	120	18	1071	1071	NUM
ejpam-4412	120	19	since	since	SCONJ
ejpam-4412	120	20	x	x	PRON
ejpam-4412	120	21	is	be	AUX
ejpam-4412	120	22	injective	injective	ADJ
ejpam-4412	120	23	and	and	CCONJ
ejpam-4412	120	24	has	have	VERB
ejpam-4412	120	25	a	a	DET
ejpam-4412	120	26	dense	dense	ADJ
ejpam-4412	120	27	range	range	NOUN
ejpam-4412	120	28	,	,	PUNCT
ejpam-4412	120	29	u11	u11	PROPN
ejpam-4412	120	30	=	=	PUNCT
ejpam-4412	120	31	w1	w1	NOUN
ejpam-4412	120	32	is	be	AUX
ejpam-4412	120	33	isometry	isometry	NOUN
ejpam-4412	120	34	and	and	CCONJ
ejpam-4412	120	35	u12z	u12z	PROPN
ejpam-4412	120	36	t	t	PROPN
ejpam-4412	120	37	=	=	PUNCT
ejpam-4412	120	38	xtw2	xtw2	PROPN
ejpam-4412	120	39	.	.	PUNCT
ejpam-4412	121	1	then	then	ADV
ejpam-4412	121	2	u∗u	u∗u	PUNCT
ejpam-4412	121	3	=	=	SYM
ejpam-4412	121	4	(	(	PUNCT
ejpam-4412	121	5	u∗	u∗	INTJ
ejpam-4412	121	6	11u11	11u11	NUM
ejpam-4412	121	7	+	+	CCONJ
ejpam-4412	121	8	u∗	u∗	ADJ
ejpam-4412	121	9	21u21	21u21	NUM
ejpam-4412	121	10	u∗	u∗	NOUN
ejpam-4412	121	11	11ul2	11ul2	NOUN
ejpam-4412	121	12	+	+	CCONJ
ejpam-4412	121	13	u∗	u∗	ADJ
ejpam-4412	121	14	21u22	21u22	NUM
ejpam-4412	121	15	u∗	u∗	NOUN
ejpam-4412	121	16	12u11	12u11	NUM
ejpam-4412	122	1	+	+	CCONJ
ejpam-4412	122	2	u∗	u∗	ADJ
ejpam-4412	122	3	22u21	22u21	NUM
ejpam-4412	122	4	u∗	u∗	NOUN
ejpam-4412	122	5	12u12	12u12	NUM
ejpam-4412	122	6	+	+	CCONJ
ejpam-4412	122	7	u∗	u∗	ADJ
ejpam-4412	122	8	22u22	22u22	NUM
ejpam-4412	122	9	)	)	PUNCT
ejpam-4412	122	10	onh	onh	PROPN
ejpam-4412	122	11	=	=	PUNCT
ejpam-4412	122	12	ran(t	ran(t	X
ejpam-4412	122	13	(	(	PUNCT
ejpam-4412	122	14	s	s	X
ejpam-4412	122	15	,	,	PUNCT
ejpam-4412	122	16	t))⊕ker((t	t))⊕ker((t	PROPN
ejpam-4412	122	17	(	(	PUNCT
ejpam-4412	122	18	s	s	PROPN
ejpam-4412	122	19	,	,	PUNCT
ejpam-4412	122	20	t))∗	t))∗	PROPN
ejpam-4412	122	21	)	)	PUNCT
ejpam-4412	122	22	is	be	AUX
ejpam-4412	122	23	the	the	DET
ejpam-4412	122	24	orthogonal	orthogonal	ADJ
ejpam-4412	122	25	projection	projection	NOUN
ejpam-4412	122	26	onto	onto	ADP
ejpam-4412	122	27	ran(|t	ran(|t	NOUN
ejpam-4412	122	28	|	|	ADV
ejpam-4412	122	29	)	)	PUNCT
ejpam-4412	122	30	⊃	⊃	PROPN
ejpam-4412	122	31	ran(t	ran(t	X
ejpam-4412	122	32	(	(	PUNCT
ejpam-4412	122	33	s	s	PROPN
ejpam-4412	122	34	,	,	PUNCT
ejpam-4412	122	35	t	t	PROPN
ejpam-4412	122	36	)	)	PUNCT
ejpam-4412	122	37	)	)	PUNCT
ejpam-4412	122	38	,	,	PUNCT
ejpam-4412	122	39	we	we	PRON
ejpam-4412	122	40	have	have	VERB
ejpam-4412	122	41	u21	u21	NOUN
ejpam-4412	122	42	=	=	SYM
ejpam-4412	122	43	0	0	NUM
ejpam-4412	122	44	and	and	CCONJ
ejpam-4412	122	45	u∗u	u∗u	SYM
ejpam-4412	122	46	=	=	SYM
ejpam-4412	122	47	(	(	PUNCT
ejpam-4412	122	48	1	1	NUM
ejpam-4412	122	49	0	0	NUM
ejpam-4412	122	50	0	0	NUM
ejpam-4412	122	51	u∗	u∗	NOUN
ejpam-4412	122	52	12u12	12u12	NUM
ejpam-4412	122	53	+	+	CCONJ
ejpam-4412	122	54	u∗	u∗	ADJ
ejpam-4412	122	55	22u22	22u22	NUM
ejpam-4412	122	56	)	)	PUNCT
ejpam-4412	122	57	.	.	PUNCT
ejpam-4412	123	1	since	since	SCONJ
ejpam-4412	123	2	u12z	u12z	PROPN
ejpam-4412	123	3	t	t	PROPN
ejpam-4412	123	4	=	=	PUNCT
ejpam-4412	123	5	xtw2	xtw2	PROPN
ejpam-4412	123	6	,	,	PUNCT
ejpam-4412	123	7	we	we	PRON
ejpam-4412	123	8	have	have	VERB
ejpam-4412	123	9	z2	z2	PROPN
ejpam-4412	123	10	t	t	PROPN
ejpam-4412	123	11	≥	≥	NOUN
ejpam-4412	123	12	ztu∗	ztu∗	NOUN
ejpam-4412	124	1	12u12z	12u12z	NUM
ejpam-4412	124	2	t	t	X
ejpam-4412	124	3	=	=	SYM
ejpam-4412	124	4	w	w	PROPN
ejpam-4412	124	5	∗	∗	NOUN
ejpam-4412	124	6	2x	2x	NUM
ejpam-4412	124	7	2tw2	2tw2	NUM
ejpam-4412	124	8	=	=	SYM
ejpam-4412	124	9	y	y	PROPN
ejpam-4412	124	10	2	2	NUM
ejpam-4412	124	11	t	t	PROPN
ejpam-4412	124	12	,	,	PUNCT
ejpam-4412	124	13	and	and	CCONJ
ejpam-4412	124	14	z2rp	z2rp	NUM
ejpam-4412	124	15	≥	≥	X
ejpam-4412	124	16	(	(	PUNCT
ejpam-4412	124	17	ztu∗	ztu∗	NOUN
ejpam-4412	124	18	12u12z	12u12z	NUM
ejpam-4412	124	19	t	t	NOUN
ejpam-4412	124	20	)	)	PUNCT
ejpam-4412	124	21	rp	rp	NOUN
ejpam-4412	124	22	t	t	PROPN
ejpam-4412	124	23	=	=	SYM
ejpam-4412	124	24	(	(	PUNCT
ejpam-4412	124	25	w	w	NOUN
ejpam-4412	124	26	∗	∗	X
ejpam-4412	124	27	2x	2x	NUM
ejpam-4412	124	28	tw2	tw2	PROPN
ejpam-4412	124	29	)	)	PUNCT
ejpam-4412	124	30	rp	rp	NOUN
ejpam-4412	124	31	t	t	NOUN
ejpam-4412	124	32	=	=	SYM
ejpam-4412	124	33	y	y	PROPN
ejpam-4412	124	34	2rp	2rp	ADJ
ejpam-4412	124	35	≥	≥	NOUN
ejpam-4412	124	36	z2rp	z2rp	PUNCT
ejpam-4412	124	37	by	by	ADP
ejpam-4412	124	38	löwner	löwner	NOUN
ejpam-4412	124	39	-	-	PUNCT
ejpam-4412	124	40	heinz	heinz	ADJ
ejpam-4412	124	41	inequality	inequality	NOUN
ejpam-4412	124	42	and	and	CCONJ
ejpam-4412	124	43	(	(	PUNCT
ejpam-4412	124	44	4	4	NUM
ejpam-4412	124	45	)	)	PUNCT
ejpam-4412	124	46	.	.	PUNCT
ejpam-4412	125	1	hence	hence	ADV
ejpam-4412	125	2	(	(	PUNCT
ejpam-4412	125	3	ztu∗	ztu∗	PROPN
ejpam-4412	125	4	12u12z	12u12z	NUM
ejpam-4412	125	5	t	t	NOUN
ejpam-4412	125	6	)	)	PUNCT
ejpam-4412	125	7	rp	rp	NOUN
ejpam-4412	125	8	t	t	PROPN
ejpam-4412	125	9	=	=	PUNCT
ejpam-4412	125	10	z2rp	z2rp	PUNCT
ejpam-4412	125	11	=	=	SYM
ejpam-4412	125	12	y	y	PROPN
ejpam-4412	125	13	2rp	2rp	ADJ
ejpam-4412	125	14	,	,	PUNCT
ejpam-4412	125	15	so	so	SCONJ
ejpam-4412	125	16	z	z	NOUN
ejpam-4412	125	17	=	=	SYM
ejpam-4412	125	18	y	y	PROPN
ejpam-4412	125	19	and	and	CCONJ
ejpam-4412	125	20	|t	|t	PROPN
ejpam-4412	125	21	(	(	PUNCT
ejpam-4412	125	22	s	s	X
ejpam-4412	125	23	,	,	PUNCT
ejpam-4412	125	24	t)|	t)|	NOUN
ejpam-4412	125	25	=	=	PUNCT
ejpam-4412	125	26	|t	|t	PROPN
ejpam-4412	125	27	|s+t	|s+t	NUM
ejpam-4412	125	28	.	.	PUNCT
ejpam-4412	126	1	since	since	SCONJ
ejpam-4412	126	2	z2	z2	PROPN
ejpam-4412	126	3	t	t	NOUN
ejpam-4412	126	4	=	=	SYM
ejpam-4412	126	5	ztu∗	ztu∗	NOUN
ejpam-4412	126	6	12u12z	12u12z	NUM
ejpam-4412	126	7	t	t	NOUN
ejpam-4412	126	8	≤	≤	NUM
ejpam-4412	126	9	ztu∗	ztu∗	NOUN
ejpam-4412	126	10	12u12z	12u12z	NUM
ejpam-4412	126	11	t	t	NOUN
ejpam-4412	127	1	+	+	CCONJ
ejpam-4412	127	2	ztu∗	ztu∗	NOUN
ejpam-4412	128	1	22u22z	22u22z	NUM
ejpam-4412	128	2	t	t	PROPN
ejpam-4412	128	3	≤	≤	NOUN
ejpam-4412	128	4	z2	z2	PROPN
ejpam-4412	128	5	t	t	NOUN
ejpam-4412	128	6	ztu∗	ztu∗	NOUN
ejpam-4412	129	1	22u22z	22u22z	NUM
ejpam-4412	129	2	t	t	NOUN
ejpam-4412	129	3	=	=	SYM
ejpam-4412	129	4	0	0	NUM
ejpam-4412	129	5	and	and	CCONJ
ejpam-4412	129	6	u22z	u22z	ADP
ejpam-4412	129	7	t	t	NOUN
ejpam-4412	129	8	=	=	SYM
ejpam-4412	129	9	0	0	PROPN
ejpam-4412	129	10	.	.	PUNCT
ejpam-4412	130	1	this	this	PRON
ejpam-4412	130	2	implies	imply	VERB
ejpam-4412	130	3	ran(u∗	ran(u∗	PUNCT
ejpam-4412	130	4	22	22	NUM
ejpam-4412	130	5	)	)	PUNCT
ejpam-4412	130	6	⊂	⊂	PROPN
ejpam-4412	130	7	ker(z	ker(z	PROPN
ejpam-4412	130	8	)	)	PUNCT
ejpam-4412	130	9	.	.	PUNCT
ejpam-4412	131	1	since	since	SCONJ
ejpam-4412	131	2	ran(u∗	ran(u∗	X
ejpam-4412	131	3	12u12	12u12	NUM
ejpam-4412	131	4	+	+	CCONJ
ejpam-4412	131	5	u∗	u∗	ADJ
ejpam-4412	131	6	22u22	22u22	NUM
ejpam-4412	131	7	)	)	PUNCT
ejpam-4412	131	8	⊂	⊂	PROPN
ejpam-4412	131	9	ran(z	ran(z	PROPN
ejpam-4412	131	10	)	)	PUNCT
ejpam-4412	131	11	and	and	CCONJ
ejpam-4412	131	12	u∗	u∗	VERB
ejpam-4412	131	13	22u22	22u22	NUM
ejpam-4412	131	14	≤	≤	NUM
ejpam-4412	132	1	u∗	u∗	ADJ
ejpam-4412	132	2	12u12	12u12	NUM
ejpam-4412	132	3	+	+	CCONJ
ejpam-4412	132	4	u∗	u∗	ADJ
ejpam-4412	132	5	22u22	22u22	NUM
ejpam-4412	132	6	,	,	PUNCT
ejpam-4412	132	7	we	we	PRON
ejpam-4412	132	8	have	have	VERB
ejpam-4412	132	9	ran(u∗	ran(u∗	NUM
ejpam-4412	132	10	22	22	NUM
ejpam-4412	132	11	)	)	PUNCT
ejpam-4412	133	1	⊂	⊂	PROPN
ejpam-4412	133	2	ran(z	ran(z	PROPN
ejpam-4412	133	3	)	)	PUNCT
ejpam-4412	133	4	.	.	PUNCT
ejpam-4412	134	1	hence	hence	ADV
ejpam-4412	134	2	u22	u22	PROPN
ejpam-4412	134	3	=	=	SYM
ejpam-4412	134	4	0	0	NUM
ejpam-4412	134	5	,	,	PUNCT
ejpam-4412	134	6	u	u	NOUN
ejpam-4412	134	7	=	=	PUNCT
ejpam-4412	134	8	(	(	PUNCT
ejpam-4412	134	9	w1	w1	NOUN
ejpam-4412	134	10	u12	u12	PROPN
ejpam-4412	134	11	0	0	NUM
ejpam-4412	134	12	0	0	NUM
ejpam-4412	134	13	)	)	PUNCT
ejpam-4412	134	14	and	and	CCONJ
ejpam-4412	134	15	ran(u	ran(u	PROPN
ejpam-4412	134	16	)	)	PUNCT
ejpam-4412	135	1	⊂	⊂	PROPN
ejpam-4412	135	2	ran(t	ran(t	X
ejpam-4412	135	3	(	(	PUNCT
ejpam-4412	135	4	s	s	PROPN
ejpam-4412	135	5	,	,	PUNCT
ejpam-4412	135	6	t	t	PROPN
ejpam-4412	135	7	)	)	PUNCT
ejpam-4412	135	8	)	)	PUNCT
ejpam-4412	136	1	⊂	⊂	PROPN
ejpam-4412	136	2	ℜ(|t	ℜ(|t	VERB
ejpam-4412	136	3	|	|	ADV
ejpam-4412	136	4	)	)	PUNCT
ejpam-4412	137	1	=	=	SYM
ejpam-4412	137	2	ran(e	ran(e	PROPN
ejpam-4412	137	3	)	)	PUNCT
ejpam-4412	137	4	.	.	PUNCT
ejpam-4412	138	1	since	since	SCONJ
ejpam-4412	138	2	w	w	NOUN
ejpam-4412	138	3	commutes	commute	NOUN
ejpam-4412	138	4	with	with	ADP
ejpam-4412	138	5	|t	|t	PROPN
ejpam-4412	138	6	(	(	PUNCT
ejpam-4412	138	7	s	s	PROPN
ejpam-4412	138	8	,	,	PUNCT
ejpam-4412	138	9	t)|	t)|	NOUN
ejpam-4412	138	10	=	=	PUNCT
ejpam-4412	138	11	|t	|t	PROPN
ejpam-4412	138	12	|s+t	|s+t	NUM
ejpam-4412	138	13	,	,	PUNCT
ejpam-4412	138	14	w	w	ADJ
ejpam-4412	138	15	commutes	commute	NOUN
ejpam-4412	138	16	with	with	ADP
ejpam-4412	138	17	|t	|t	PROPN
ejpam-4412	138	18	|	|	ADV
ejpam-4412	138	19	and	and	CCONJ
ejpam-4412	138	20	|t	|t	VERB
ejpam-4412	138	21	|s(w	|s(w	PROPN
ejpam-4412	138	22	−	−	PROPN
ejpam-4412	138	23	u)|t	u)|t	NOUN
ejpam-4412	138	24	|t	|t	PROPN
ejpam-4412	139	1	=	=	PROPN
ejpam-4412	140	1	w	w	X
ejpam-4412	140	2	|t	|t	PROPN
ejpam-4412	140	3	|s|t	|s|t	VERB
ejpam-4412	140	4	|t	|t	PROPN
ejpam-4412	141	1	−	−	PROPN
ejpam-4412	141	2	|t	|t	PROPN
ejpam-4412	142	1	|su	|su	NOUN
ejpam-4412	142	2	|t	|t	VERB
ejpam-4412	142	3	|t	|t	PROPN
ejpam-4412	142	4	=	=	SYM
ejpam-4412	143	1	w	w	X
ejpam-4412	143	2	|t	|t	PROPN
ejpam-4412	143	3	(	(	PUNCT
ejpam-4412	143	4	s	s	X
ejpam-4412	143	5	,	,	PUNCT
ejpam-4412	143	6	t)|	t)|	ADV
ejpam-4412	143	7	−	−	PROPN
ejpam-4412	143	8	t	t	PROPN
ejpam-4412	143	9	(	(	PUNCT
ejpam-4412	143	10	s	s	PROPN
ejpam-4412	143	11	,	,	PUNCT
ejpam-4412	143	12	t	t	PROPN
ejpam-4412	143	13	)	)	PUNCT
ejpam-4412	143	14	=	=	SYM
ejpam-4412	144	1	0	0	X
ejpam-4412	144	2	.	.	PUNCT
ejpam-4412	145	1	hence	hence	ADV
ejpam-4412	145	2	e(w	e(w	VERB
ejpam-4412	145	3	−	−	NOUN
ejpam-4412	145	4	u)e	u)e	ADJ
ejpam-4412	145	5	=	=	SYM
ejpam-4412	145	6	0	0	NUM
ejpam-4412	145	7	and	and	CCONJ
ejpam-4412	145	8	u	u	X
ejpam-4412	145	9	=	=	PROPN
ejpam-4412	145	10	ue	ue	PROPN
ejpam-4412	145	11	=	=	PROPN
ejpam-4412	145	12	eue	eue	PROPN
ejpam-4412	145	13	=	=	SYM
ejpam-4412	145	14	ewe	ewe	PROPN
ejpam-4412	146	1	=	=	NOUN
ejpam-4412	146	2	we	we	PRON
ejpam-4412	146	3	=	=	PUNCT
ejpam-4412	146	4	w.	w.	NOUN
ejpam-4412	146	5	thus	thus	ADV
ejpam-4412	146	6	u	u	X
ejpam-4412	146	7	=	=	SYM
ejpam-4412	146	8	w	w	NOUN
ejpam-4412	146	9	commutes	commute	NOUN
ejpam-4412	146	10	with	with	ADP
ejpam-4412	146	11	|t	|t	PROPN
ejpam-4412	147	1	|	|	ADV
ejpam-4412	147	2	and	and	CCONJ
ejpam-4412	147	3	t	t	PROPN
ejpam-4412	147	4	is	be	AUX
ejpam-4412	147	5	quasinormal	quasinormal	ADJ
ejpam-4412	147	6	.	.	PUNCT
ejpam-4412	148	1	corollary	corollary	ADJ
ejpam-4412	148	2	1	1	NUM
ejpam-4412	148	3	.	.	PUNCT
ejpam-4412	149	1	let	let	AUX
ejpam-4412	149	2	t	t	NOUN
ejpam-4412	149	3	=	=	SYM
ejpam-4412	149	4	u	u	NOUN
ejpam-4412	149	5	|t	|t	VERB
ejpam-4412	149	6	|	|	ADV
ejpam-4412	149	7	be	be	AUX
ejpam-4412	149	8	a	a	DET
ejpam-4412	149	9	class	class	NOUN
ejpam-4412	149	10	p	p	NOUN
ejpam-4412	149	11	-	-	PUNCT
ejpam-4412	149	12	wa(s	wa(s	NUM
ejpam-4412	149	13	,	,	PUNCT
ejpam-4412	149	14	t	t	NOUN
ejpam-4412	149	15	)	)	PUNCT
ejpam-4412	149	16	operator	operator	NOUN
ejpam-4412	149	17	.	.	PUNCT
ejpam-4412	150	1	if	if	SCONJ
ejpam-4412	150	2	t	t	PROPN
ejpam-4412	150	3	(	(	PUNCT
ejpam-4412	150	4	s	s	PROPN
ejpam-4412	150	5	,	,	PUNCT
ejpam-4412	150	6	t	t	PROPN
ejpam-4412	150	7	)	)	PUNCT
ejpam-4412	150	8	=	=	PUNCT
ejpam-4412	150	9	|t	|t	PROPN
ejpam-4412	150	10	|su	|su	NOUN
ejpam-4412	150	11	|t	|t	VERB
ejpam-4412	150	12	|t	|t	VERB
ejpam-4412	150	13	is	be	AUX
ejpam-4412	150	14	normal	normal	ADJ
ejpam-4412	150	15	,	,	PUNCT
ejpam-4412	150	16	then	then	ADV
ejpam-4412	150	17	t	t	PROPN
ejpam-4412	150	18	is	be	AUX
ejpam-4412	150	19	also	also	ADV
ejpam-4412	150	20	normal	normal	ADJ
ejpam-4412	150	21	.	.	PUNCT
ejpam-4412	151	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	151	2	,	,	PUNCT
ejpam-4412	151	3	n.	n.	NOUN
ejpam-4412	151	4	h.	h.	PROPN
ejpam-4412	151	5	altaweel	altaweel	PROPN
ejpam-4412	151	6	/	/	SYM
ejpam-4412	151	7	eur	eur	PROPN
ejpam-4412	151	8	.	.	PUNCT
ejpam-4412	152	1	j.	j.	PROPN
ejpam-4412	152	2	pure	pure	PROPN
ejpam-4412	152	3	appl	appl	PROPN
ejpam-4412	152	4	.	.	PROPN
ejpam-4412	152	5	math	math	PROPN
ejpam-4412	152	6	,	,	PUNCT
ejpam-4412	152	7	15	15	NUM
ejpam-4412	152	8	(	(	PUNCT
ejpam-4412	152	9	3	3	NUM
ejpam-4412	152	10	)	)	PUNCT
ejpam-4412	152	11	(	(	PUNCT
ejpam-4412	152	12	2022	2022	NUM
ejpam-4412	152	13	)	)	PUNCT
ejpam-4412	152	14	,	,	PUNCT
ejpam-4412	152	15	1067	1067	NUM
ejpam-4412	152	16	-	-	SYM
ejpam-4412	152	17	1089	1089	NUM
ejpam-4412	152	18	1072	1072	NUM
ejpam-4412	152	19	proof	proof	NOUN
ejpam-4412	152	20	.	.	PUNCT
ejpam-4412	153	1	since	since	SCONJ
ejpam-4412	153	2	t	t	PROPN
ejpam-4412	153	3	(	(	PUNCT
ejpam-4412	153	4	s	s	PROPN
ejpam-4412	153	5	,	,	PUNCT
ejpam-4412	153	6	t	t	PROPN
ejpam-4412	153	7	)	)	PUNCT
ejpam-4412	153	8	is	be	AUX
ejpam-4412	153	9	normal	normal	ADJ
ejpam-4412	153	10	,	,	PUNCT
ejpam-4412	153	11	t	t	PROPN
ejpam-4412	153	12	is	be	AUX
ejpam-4412	153	13	quasinormal	quasinormal	ADJ
ejpam-4412	153	14	by	by	ADP
ejpam-4412	153	15	theorem	theorem	NOUN
ejpam-4412	153	16	1	1	NUM
ejpam-4412	153	17	.	.	PUNCT
ejpam-4412	154	1	hence	hence	ADV
ejpam-4412	154	2	t	t	PROPN
ejpam-4412	154	3	(	(	PUNCT
ejpam-4412	154	4	s	s	PROPN
ejpam-4412	154	5	,	,	PUNCT
ejpam-4412	154	6	t	t	PROPN
ejpam-4412	154	7	)	)	PUNCT
ejpam-4412	154	8	=	=	PUNCT
ejpam-4412	154	9	|t	|t	PROPN
ejpam-4412	154	10	|su	|su	NOUN
ejpam-4412	154	11	|t	|t	VERB
ejpam-4412	154	12	|t	|t	VERB
ejpam-4412	155	1	=	=	SYM
ejpam-4412	155	2	u	u	PROPN
ejpam-4412	155	3	|t	|t	PROPN
ejpam-4412	155	4	|s+t	|s+t	NUM
ejpam-4412	155	5	and	and	CCONJ
ejpam-4412	155	6	(	(	PUNCT
ejpam-4412	155	7	t	t	PROPN
ejpam-4412	155	8	(	(	PUNCT
ejpam-4412	155	9	s	s	X
ejpam-4412	155	10	,	,	PUNCT
ejpam-4412	155	11	t))∗	t))∗	PROPN
ejpam-4412	155	12	=	=	PUNCT
ejpam-4412	155	13	|t	|t	NOUN
ejpam-4412	155	14	|s+tu∗.	|s+tu∗.	PROPN
ejpam-4412	155	15	hence	hence	ADV
ejpam-4412	155	16	|t	|t	PROPN
ejpam-4412	155	17	|2(s+t	|2(s+t	NUM
ejpam-4412	155	18	)	)	PUNCT
ejpam-4412	156	1	=	=	PRON
ejpam-4412	156	2	|t	|t	PROPN
ejpam-4412	156	3	(	(	PUNCT
ejpam-4412	156	4	s	s	PROPN
ejpam-4412	156	5	,	,	PUNCT
ejpam-4412	156	6	t)|2	t)|2	NOUN
ejpam-4412	156	7	=	=	SYM
ejpam-4412	156	8	|(t	|(t	PROPN
ejpam-4412	156	9	(	(	PUNCT
ejpam-4412	156	10	s	s	PROPN
ejpam-4412	156	11	,	,	PUNCT
ejpam-4412	156	12	t))∗|2	t))∗|2	PRON
ejpam-4412	156	13	=	=	PUNCT
ejpam-4412	156	14	|t	|t	PROPN
ejpam-4412	156	15	∗|2(s+t	∗|2(s+t	NOUN
ejpam-4412	156	16	)	)	PUNCT
ejpam-4412	156	17	.	.	PUNCT
ejpam-4412	157	1	this	this	PRON
ejpam-4412	157	2	implies	imply	VERB
ejpam-4412	157	3	|t	|t	VERB
ejpam-4412	158	1	|	|	ADV
ejpam-4412	158	2	=	=	PUNCT
ejpam-4412	158	3	|t	|t	NOUN
ejpam-4412	159	1	∗|	∗|	PROPN
ejpam-4412	159	2	and	and	CCONJ
ejpam-4412	159	3	t	t	PROPN
ejpam-4412	159	4	is	be	AUX
ejpam-4412	159	5	normal	normal	ADJ
ejpam-4412	159	6	.	.	PUNCT
ejpam-4412	160	1	theorem	theorem	NOUN
ejpam-4412	160	2	2	2	NUM
ejpam-4412	160	3	.	.	PUNCT
ejpam-4412	161	1	[	[	X
ejpam-4412	161	2	25	25	NUM
ejpam-4412	161	3	]	]	PUNCT
ejpam-4412	161	4	let	let	VERB
ejpam-4412	161	5	s1	s1	PROPN
ejpam-4412	161	6	>	>	X
ejpam-4412	161	7	0	0	PROPN
ejpam-4412	161	8	,	,	PUNCT
ejpam-4412	161	9	s2	s2	VERB
ejpam-4412	161	10	>	>	X
ejpam-4412	161	11	0	0	PROPN
ejpam-4412	161	12	,	,	PUNCT
ejpam-4412	161	13	t1	t1	NOUN
ejpam-4412	161	14	>	>	X
ejpam-4412	161	15	0	0	PROPN
ejpam-4412	161	16	,	,	PUNCT
ejpam-4412	161	17	t2	t2	NOUN
ejpam-4412	161	18	>	>	X
ejpam-4412	161	19	0	0	PUNCT
ejpam-4412	162	1	and	and	CCONJ
ejpam-4412	162	2	0	0	NUM
ejpam-4412	162	3	<	<	X
ejpam-4412	162	4	p	p	X
ejpam-4412	162	5	≤	≤	NUM
ejpam-4412	162	6	1	1	NUM
ejpam-4412	162	7	.	.	PUNCT
ejpam-4412	163	1	if	if	SCONJ
ejpam-4412	163	2	t	t	PROPN
ejpam-4412	163	3	belongs	belong	VERB
ejpam-4412	163	4	to	to	ADP
ejpam-4412	163	5	class	class	NOUN
ejpam-4412	163	6	p1	p1	NOUN
ejpam-4412	163	7	-	-	PUNCT
ejpam-4412	163	8	wa(s1	wa(s1	PROPN
ejpam-4412	163	9	,	,	PUNCT
ejpam-4412	163	10	t1	t1	NOUN
ejpam-4412	163	11	)	)	PUNCT
ejpam-4412	163	12	for	for	ADP
ejpam-4412	163	13	0	0	NUM
ejpam-4412	163	14	<	<	X
ejpam-4412	163	15	p1	p1	PROPN
ejpam-4412	163	16	≤	≤	PUNCT
ejpam-4412	163	17	p	p	PROPN
ejpam-4412	163	18	and	and	CCONJ
ejpam-4412	163	19	t	t	PROPN
ejpam-4412	163	20	∗	∗	NOUN
ejpam-4412	163	21	belongs	belong	VERB
ejpam-4412	163	22	to	to	ADP
ejpam-4412	163	23	class	class	NOUN
ejpam-4412	163	24	p2	p2	PROPN
ejpam-4412	163	25	-	-	PUNCT
ejpam-4412	163	26	wa(s2	wa(s2	PROPN
ejpam-4412	163	27	,	,	PUNCT
ejpam-4412	163	28	t2	t2	NOUN
ejpam-4412	163	29	)	)	PUNCT
ejpam-4412	163	30	for	for	ADP
ejpam-4412	163	31	0	0	NUM
ejpam-4412	163	32	<	<	X
ejpam-4412	163	33	p2	p2	PROPN
ejpam-4412	163	34	≤	≤	PROPN
ejpam-4412	163	35	p	p	NOUN
ejpam-4412	163	36	,	,	PUNCT
ejpam-4412	163	37	then	then	ADV
ejpam-4412	163	38	t	t	PROPN
ejpam-4412	163	39	is	be	AUX
ejpam-4412	163	40	normal	normal	ADJ
ejpam-4412	163	41	.	.	PUNCT
ejpam-4412	164	1	to	to	PART
ejpam-4412	164	2	prove	prove	VERB
ejpam-4412	164	3	theorem	theorem	ADJ
ejpam-4412	164	4	2	2	NUM
ejpam-4412	164	5	,	,	PUNCT
ejpam-4412	164	6	we	we	PRON
ejpam-4412	164	7	need	need	VERB
ejpam-4412	164	8	the	the	DET
ejpam-4412	164	9	following	follow	VERB
ejpam-4412	164	10	results	result	NOUN
ejpam-4412	164	11	.	.	PUNCT
ejpam-4412	165	1	lemma	lemma	PROPN
ejpam-4412	165	2	1	1	NUM
ejpam-4412	165	3	.	.	PUNCT
ejpam-4412	166	1	(	(	PUNCT
ejpam-4412	166	2	[	[	X
ejpam-4412	166	3	21	21	NUM
ejpam-4412	166	4	]	]	PUNCT
ejpam-4412	166	5	)	)	PUNCT
ejpam-4412	166	6	if	if	SCONJ
ejpam-4412	166	7	t	t	PROPN
ejpam-4412	166	8	is	be	AUX
ejpam-4412	166	9	class	class	NOUN
ejpam-4412	166	10	p	p	NOUN
ejpam-4412	166	11	-	-	PUNCT
ejpam-4412	166	12	wa(s	wa(s	NUM
ejpam-4412	166	13	,	,	PUNCT
ejpam-4412	166	14	t	t	PROPN
ejpam-4412	166	15	)	)	PUNCT
ejpam-4412	166	16	and	and	CCONJ
ejpam-4412	166	17	0	0	NUM
ejpam-4412	166	18	<	<	X
ejpam-4412	166	19	s	s	PART
ejpam-4412	166	20	≤	≤	NUM
ejpam-4412	166	21	s1	s1	NOUN
ejpam-4412	166	22	,	,	PUNCT
ejpam-4412	166	23	0	0	NUM
ejpam-4412	166	24	<	<	X
ejpam-4412	166	25	t	t	X
ejpam-4412	166	26	≤	≤	NUM
ejpam-4412	166	27	t1	t1	PROPN
ejpam-4412	166	28	,	,	PUNCT
ejpam-4412	166	29	0	0	PUNCT
ejpam-4412	166	30	<	<	X
ejpam-4412	166	31	p1	p1	PROPN
ejpam-4412	166	32	≤	≤	PUNCT
ejpam-4412	167	1	p	p	X
ejpam-4412	167	2	<	<	X
ejpam-4412	167	3	1	1	NUM
ejpam-4412	167	4	,	,	PUNCT
ejpam-4412	167	5	then	then	ADV
ejpam-4412	167	6	t	t	PROPN
ejpam-4412	167	7	is	be	AUX
ejpam-4412	167	8	class	class	NOUN
ejpam-4412	167	9	p1	p1	NOUN
ejpam-4412	167	10	-	-	PUNCT
ejpam-4412	167	11	wa(s1	wa(s1	PROPN
ejpam-4412	167	12	,	,	PUNCT
ejpam-4412	167	13	t1	t1	NOUN
ejpam-4412	167	14	)	)	PUNCT
ejpam-4412	167	15	.	.	PUNCT
ejpam-4412	168	1	theorem	theorem	ADJ
ejpam-4412	168	2	3	3	NUM
ejpam-4412	168	3	(	(	PUNCT
ejpam-4412	168	4	furuta	furuta	PROPN
ejpam-4412	168	5	theorem	theorem	NOUN
ejpam-4412	168	6	[	[	X
ejpam-4412	168	7	14	14	NUM
ejpam-4412	168	8	]	]	PUNCT
ejpam-4412	168	9	)	)	PUNCT
ejpam-4412	168	10	.	.	PUNCT
ejpam-4412	169	1	if	if	SCONJ
ejpam-4412	169	2	a	a	DET
ejpam-4412	169	3	≥	≥	NOUN
ejpam-4412	169	4	b	b	NOUN
ejpam-4412	169	5	≥	≥	NOUN
ejpam-4412	169	6	0	0	NUM
ejpam-4412	169	7	,	,	PUNCT
ejpam-4412	169	8	then	then	ADV
ejpam-4412	169	9	for	for	ADP
ejpam-4412	169	10	each	each	DET
ejpam-4412	169	11	r	r	NOUN
ejpam-4412	169	12	≥	≥	NOUN
ejpam-4412	169	13	0	0	NUM
ejpam-4412	169	14	,	,	PUNCT
ejpam-4412	169	15	(	(	PUNCT
ejpam-4412	169	16	i	i	NOUN
ejpam-4412	169	17	)	)	PUNCT
ejpam-4412	169	18	(	(	PUNCT
ejpam-4412	169	19	b	b	X
ejpam-4412	169	20	r	r	NOUN
ejpam-4412	169	21	2apb	2apb	NUM
ejpam-4412	169	22	r	r	NOUN
ejpam-4412	169	23	2	2	NUM
ejpam-4412	169	24	)	)	PUNCT
ejpam-4412	169	25	1	1	NUM
ejpam-4412	169	26	q	q	NOUN
ejpam-4412	169	27	≥	≥	PROPN
ejpam-4412	169	28	b	b	PROPN
ejpam-4412	169	29	r+p	r+p	PROPN
ejpam-4412	169	30	q	q	PROPN
ejpam-4412	169	31	and	and	CCONJ
ejpam-4412	169	32	(	(	PUNCT
ejpam-4412	169	33	ii	ii	NOUN
ejpam-4412	169	34	)	)	PUNCT
ejpam-4412	169	35	a	a	DET
ejpam-4412	169	36	r+p	r+p	ADJ
ejpam-4412	169	37	q	q	X
ejpam-4412	169	38	≥	≥	X
ejpam-4412	169	39	(	(	PUNCT
ejpam-4412	169	40	a	a	DET
ejpam-4412	169	41	r	r	NOUN
ejpam-4412	169	42	2bpa	2bpa	NUM
ejpam-4412	169	43	r	r	NOUN
ejpam-4412	169	44	2	2	NUM
ejpam-4412	169	45	)	)	PUNCT
ejpam-4412	169	46	1	1	NUM
ejpam-4412	169	47	q	q	NOUN
ejpam-4412	169	48	hold	hold	VERB
ejpam-4412	169	49	for	for	ADP
ejpam-4412	169	50	p	p	PRON
ejpam-4412	169	51	≥	≥	NOUN
ejpam-4412	169	52	0	0	NUM
ejpam-4412	169	53	and	and	CCONJ
ejpam-4412	169	54	q	q	ADJ
ejpam-4412	169	55	≥	≥	NOUN
ejpam-4412	169	56	1	1	NUM
ejpam-4412	169	57	with	with	ADP
ejpam-4412	169	58	(	(	PUNCT
ejpam-4412	169	59	1	1	NUM
ejpam-4412	169	60	+	+	NUM
ejpam-4412	169	61	r)q	r)q	NOUN
ejpam-4412	169	62	≥	≥	PROPN
ejpam-4412	169	63	p+	p+	PROPN
ejpam-4412	169	64	r.	r.	PROPN
ejpam-4412	169	65	proposition	proposition	PROPN
ejpam-4412	169	66	1	1	NUM
ejpam-4412	169	67	.	.	PUNCT
ejpam-4412	170	1	(	(	PUNCT
ejpam-4412	170	2	[	[	X
ejpam-4412	170	3	19	19	NUM
ejpam-4412	170	4	]	]	PUNCT
ejpam-4412	170	5	)	)	PUNCT
ejpam-4412	170	6	let	let	VERB
ejpam-4412	170	7	a	a	DET
ejpam-4412	170	8	≥	≥	NOUN
ejpam-4412	170	9	0	0	NUM
ejpam-4412	170	10	and	and	CCONJ
ejpam-4412	170	11	b	b	NOUN
ejpam-4412	170	12	≥	≥	NOUN
ejpam-4412	170	13	0	0	NUM
ejpam-4412	170	14	.	.	PUNCT
ejpam-4412	171	1	if	if	SCONJ
ejpam-4412	171	2	b	b	NOUN
ejpam-4412	171	3	1	1	NUM
ejpam-4412	171	4	2ab	2ab	NOUN
ejpam-4412	171	5	1	1	NUM
ejpam-4412	171	6	2	2	NUM
ejpam-4412	171	7	≥	≥	NOUN
ejpam-4412	171	8	b2	b2	NOUN
ejpam-4412	171	9	and	and	CCONJ
ejpam-4412	171	10	a	a	DET
ejpam-4412	171	11	1	1	NUM
ejpam-4412	171	12	2ba	2ba	NOUN
ejpam-4412	171	13	1	1	NUM
ejpam-4412	171	14	2	2	NUM
ejpam-4412	171	15	≥	≥	NOUN
ejpam-4412	171	16	a2	a2	NOUN
ejpam-4412	171	17	,	,	PUNCT
ejpam-4412	171	18	(	(	PUNCT
ejpam-4412	171	19	5	5	NUM
ejpam-4412	171	20	)	)	PUNCT
ejpam-4412	171	21	then	then	ADV
ejpam-4412	171	22	a	a	DET
ejpam-4412	171	23	=	=	X
ejpam-4412	171	24	b.	b.	NOUN
ejpam-4412	171	25	proof	proof	NOUN
ejpam-4412	171	26	.	.	PUNCT
ejpam-4412	172	1	[	[	X
ejpam-4412	172	2	proof	proof	NOUN
ejpam-4412	172	3	of	of	ADP
ejpam-4412	172	4	theorem	theorem	NOUN
ejpam-4412	172	5	2	2	NUM
ejpam-4412	172	6	]	]	PUNCT
ejpam-4412	172	7	let	let	VERB
ejpam-4412	172	8	r	r	NOUN
ejpam-4412	172	9	=	=	SYM
ejpam-4412	172	10	max{s1	max{s1	NOUN
ejpam-4412	172	11	,	,	PUNCT
ejpam-4412	172	12	s2	s2	PROPN
ejpam-4412	172	13	,	,	PUNCT
ejpam-4412	172	14	t1	t1	NOUN
ejpam-4412	172	15	,	,	PUNCT
ejpam-4412	172	16	t2	t2	NOUN
ejpam-4412	172	17	}	}	PUNCT
ejpam-4412	172	18	and	and	CCONJ
ejpam-4412	172	19	let	let	VERB
ejpam-4412	172	20	q	q	NOUN
ejpam-4412	172	21	=	=	SYM
ejpam-4412	172	22	min{p1	min{p1	ADJ
ejpam-4412	172	23	,	,	PUNCT
ejpam-4412	172	24	p2	p2	PROPN
ejpam-4412	172	25	}	}	PUNCT
ejpam-4412	172	26	.	.	PUNCT
ejpam-4412	173	1	firstly	firstly	ADV
ejpam-4412	173	2	,	,	PUNCT
ejpam-4412	173	3	if	if	SCONJ
ejpam-4412	173	4	t	t	PROPN
ejpam-4412	173	5	belongs	belong	VERB
ejpam-4412	173	6	to	to	ADP
ejpam-4412	173	7	class	class	NOUN
ejpam-4412	173	8	p1	p1	NOUN
ejpam-4412	173	9	-	-	PUNCT
ejpam-4412	173	10	wa(s1	wa(s1	PROPN
ejpam-4412	173	11	,	,	PUNCT
ejpam-4412	173	12	t1	t1	NOUN
ejpam-4412	173	13	)	)	PUNCT
ejpam-4412	173	14	,	,	PUNCT
ejpam-4412	173	15	then	then	ADV
ejpam-4412	173	16	t	t	PROPN
ejpam-4412	173	17	belongs	belong	VERB
ejpam-4412	173	18	to	to	ADP
ejpam-4412	173	19	class	class	NOUN
ejpam-4412	173	20	q	q	NOUN
ejpam-4412	173	21	-	-	NOUN
ejpam-4412	173	22	wa(r	wa(r	NOUN
ejpam-4412	173	23	,	,	PUNCT
ejpam-4412	173	24	r	r	NOUN
ejpam-4412	173	25	)	)	PUNCT
ejpam-4412	173	26	by	by	ADP
ejpam-4412	173	27	lemma	lemma	PROPN
ejpam-4412	173	28	1	1	NUM
ejpam-4412	173	29	.	.	PUNCT
ejpam-4412	174	1	hence	hence	ADV
ejpam-4412	174	2	we	we	PRON
ejpam-4412	174	3	have	have	VERB
ejpam-4412	174	4	(	(	PUNCT
ejpam-4412	174	5	|t	|t	NOUN
ejpam-4412	174	6	∗|r|t	∗|r|t	NUM
ejpam-4412	174	7	|2r|t	|2r|t	ADJ
ejpam-4412	174	8	∗|r	∗|r	NOUN
ejpam-4412	174	9	)	)	PUNCT
ejpam-4412	174	10	q	q	NOUN
ejpam-4412	174	11	2	2	NUM
ejpam-4412	174	12	≥	≥	NOUN
ejpam-4412	174	13	|t	|t	VERB
ejpam-4412	174	14	∗|2rq	∗|2rq	NUM
ejpam-4412	174	15	and	and	CCONJ
ejpam-4412	174	16	|t	|t	VERB
ejpam-4412	175	1	|2rq	|2rq	PROPN
ejpam-4412	175	2	≥	≥	NUM
ejpam-4412	175	3	(	(	PUNCT
ejpam-4412	175	4	|t	|t	VERB
ejpam-4412	175	5	|r|t	|r|t	CCONJ
ejpam-4412	175	6	∗|2r|t	∗|2r|t	PROPN
ejpam-4412	175	7	|r	|r	PROPN
ejpam-4412	175	8	)	)	PUNCT
ejpam-4412	175	9	q	q	NOUN
ejpam-4412	175	10	2	2	NUM
ejpam-4412	175	11	(	(	PUNCT
ejpam-4412	175	12	6	6	NUM
ejpam-4412	175	13	)	)	PUNCT
ejpam-4412	175	14	secondly	secondly	ADV
ejpam-4412	175	15	,	,	PUNCT
ejpam-4412	175	16	if	if	SCONJ
ejpam-4412	175	17	t	t	PROPN
ejpam-4412	175	18	∗	∗	NOUN
ejpam-4412	175	19	belongs	belong	VERB
ejpam-4412	175	20	to	to	ADP
ejpam-4412	175	21	class	class	NOUN
ejpam-4412	175	22	p2	p2	PROPN
ejpam-4412	175	23	-	-	PUNCT
ejpam-4412	175	24	wa(s2	wa(s2	PROPN
ejpam-4412	175	25	,	,	PUNCT
ejpam-4412	175	26	t2	t2	NOUN
ejpam-4412	175	27	)	)	PUNCT
ejpam-4412	175	28	,	,	PUNCT
ejpam-4412	175	29	then	then	ADV
ejpam-4412	175	30	t	t	PROPN
ejpam-4412	175	31	∗	∗	NOUN
ejpam-4412	175	32	belongs	belong	VERB
ejpam-4412	175	33	to	to	ADP
ejpam-4412	175	34	class	class	NOUN
ejpam-4412	175	35	q	q	NOUN
ejpam-4412	175	36	-	-	NOUN
ejpam-4412	175	37	wa(r	wa(r	NOUN
ejpam-4412	175	38	,	,	PUNCT
ejpam-4412	175	39	r	r	NOUN
ejpam-4412	175	40	)	)	PUNCT
ejpam-4412	175	41	by	by	ADP
ejpam-4412	175	42	lemma	lemma	PROPN
ejpam-4412	175	43	1	1	NUM
ejpam-4412	175	44	.	.	PUNCT
ejpam-4412	176	1	hence	hence	ADV
ejpam-4412	176	2	we	we	PRON
ejpam-4412	176	3	have	have	VERB
ejpam-4412	176	4	(	(	PUNCT
ejpam-4412	176	5	|t	|t	VERB
ejpam-4412	176	6	|r|t	|r|t	CCONJ
ejpam-4412	176	7	∗|2r|t	∗|2r|t	PROPN
ejpam-4412	176	8	|r	|r	PROPN
ejpam-4412	176	9	)	)	PUNCT
ejpam-4412	176	10	q	q	PROPN
ejpam-4412	176	11	2	2	NUM
ejpam-4412	176	12	≥	≥	NOUN
ejpam-4412	176	13	|t	|t	VERB
ejpam-4412	177	1	|2rq	|2rq	ADJ
ejpam-4412	177	2	and	and	CCONJ
ejpam-4412	177	3	|t	|t	VERB
ejpam-4412	177	4	∗|2rq	∗|2rq	NUM
ejpam-4412	177	5	≥	≥	NOUN
ejpam-4412	177	6	(	(	PUNCT
ejpam-4412	177	7	|t	|t	NOUN
ejpam-4412	177	8	∗|r|t	∗|r|t	NUM
ejpam-4412	177	9	|2r|t	|2r|t	ADJ
ejpam-4412	177	10	∗|r	∗|r	NOUN
ejpam-4412	177	11	)	)	PUNCT
ejpam-4412	177	12	q	q	NOUN
ejpam-4412	177	13	2	2	NUM
ejpam-4412	177	14	(	(	PUNCT
ejpam-4412	177	15	7	7	NUM
ejpam-4412	177	16	)	)	PUNCT
ejpam-4412	177	17	therefore	therefore	ADV
ejpam-4412	177	18	|t	|t	PROPN
ejpam-4412	177	19	∗|r|t	∗|r|t	NUM
ejpam-4412	177	20	|2r|t	|2r|t	ADJ
ejpam-4412	177	21	∗|r	∗|r	NOUN
ejpam-4412	177	22	=	=	SYM
ejpam-4412	177	23	|t	|t	PROPN
ejpam-4412	177	24	∗|4r	∗|4r	NOUN
ejpam-4412	177	25	and	and	CCONJ
ejpam-4412	177	26	|t	|t	PROPN
ejpam-4412	177	27	|4r	|4r	X
ejpam-4412	177	28	=	=	PUNCT
ejpam-4412	177	29	|t	|t	VERB
ejpam-4412	177	30	|r|t	|r|t	CCONJ
ejpam-4412	178	1	∗|2r|t	∗|2r|t	ADJ
ejpam-4412	178	2	|r	|r	NOUN
ejpam-4412	178	3	hold	hold	VERB
ejpam-4412	178	4	by	by	ADP
ejpam-4412	178	5	(	(	PUNCT
ejpam-4412	178	6	6	6	NUM
ejpam-4412	178	7	)	)	PUNCT
ejpam-4412	178	8	and	and	CCONJ
ejpam-4412	178	9	(	(	PUNCT
ejpam-4412	178	10	7	7	NUM
ejpam-4412	178	11	)	)	PUNCT
ejpam-4412	178	12	,	,	PUNCT
ejpam-4412	178	13	and	and	CCONJ
ejpam-4412	178	14	then	then	ADV
ejpam-4412	178	15	|t	|t	VERB
ejpam-4412	179	1	|	|	ADV
ejpam-4412	179	2	=	=	PUNCT
ejpam-4412	179	3	|t	|t	NOUN
ejpam-4412	179	4	∗|	∗|	NOUN
ejpam-4412	179	5	by	by	ADP
ejpam-4412	179	6	proposition	proposition	NOUN
ejpam-4412	179	7	1	1	NUM
ejpam-4412	179	8	.	.	PUNCT
ejpam-4412	180	1	the	the	DET
ejpam-4412	180	2	following	following	ADJ
ejpam-4412	180	3	result	result	NOUN
ejpam-4412	180	4	is	be	AUX
ejpam-4412	180	5	very	very	ADV
ejpam-4412	180	6	important	important	ADJ
ejpam-4412	180	7	in	in	ADP
ejpam-4412	180	8	the	the	DET
ejpam-4412	180	9	sequal	sequal	ADJ
ejpam-4412	180	10	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	180	11	,	,	PUNCT
ejpam-4412	180	12	n.	n.	NOUN
ejpam-4412	180	13	h.	h.	PROPN
ejpam-4412	180	14	altaweel	altaweel	PROPN
ejpam-4412	180	15	/	/	SYM
ejpam-4412	180	16	eur	eur	PROPN
ejpam-4412	180	17	.	.	PUNCT
ejpam-4412	181	1	j.	j.	PROPN
ejpam-4412	181	2	pure	pure	PROPN
ejpam-4412	181	3	appl	appl	PROPN
ejpam-4412	181	4	.	.	PROPN
ejpam-4412	181	5	math	math	PROPN
ejpam-4412	181	6	,	,	PUNCT
ejpam-4412	181	7	15	15	NUM
ejpam-4412	181	8	(	(	PUNCT
ejpam-4412	181	9	3	3	NUM
ejpam-4412	181	10	)	)	PUNCT
ejpam-4412	181	11	(	(	PUNCT
ejpam-4412	181	12	2022	2022	NUM
ejpam-4412	181	13	)	)	PUNCT
ejpam-4412	181	14	,	,	PUNCT
ejpam-4412	181	15	1067	1067	NUM
ejpam-4412	181	16	-	-	SYM
ejpam-4412	181	17	1089	1089	NUM
ejpam-4412	181	18	1073	1073	NUM
ejpam-4412	181	19	theorem	theorem	NOUN
ejpam-4412	181	20	4	4	NUM
ejpam-4412	181	21	.	.	PUNCT
ejpam-4412	182	1	[	[	X
ejpam-4412	182	2	17	17	NUM
ejpam-4412	182	3	,	,	PUNCT
ejpam-4412	182	4	jensen	jensen	PROPN
ejpam-4412	182	5	’s	’s	PART
ejpam-4412	182	6	operator	operator	NOUN
ejpam-4412	182	7	inequality	inequality	NOUN
ejpam-4412	182	8	(	(	PUNCT
ejpam-4412	182	9	joi	joi	PROPN
ejpam-4412	182	10	)	)	PUNCT
ejpam-4412	182	11	]	]	PUNCT
ejpam-4412	182	12	suppose	suppose	VERB
ejpam-4412	182	13	that	that	SCONJ
ejpam-4412	182	14	f	f	PROPN
ejpam-4412	182	15	is	be	AUX
ejpam-4412	182	16	a	a	DET
ejpam-4412	182	17	continuous	continuous	ADJ
ejpam-4412	182	18	function	function	NOUN
ejpam-4412	182	19	defined	define	VERB
ejpam-4412	182	20	on	on	ADP
ejpam-4412	182	21	an	an	DET
ejpam-4412	182	22	interval	interval	NOUN
ejpam-4412	182	23	i.	i.	NOUN
ejpam-4412	182	24	then	then	ADV
ejpam-4412	182	25	f	f	PROPN
ejpam-4412	182	26	is	be	AUX
ejpam-4412	182	27	operator	operator	NOUN
ejpam-4412	182	28	convex	convex	NOUN
ejpam-4412	182	29	on	on	ADP
ejpam-4412	182	30	an	an	DET
ejpam-4412	182	31	interval	interval	NOUN
ejpam-4412	182	32	i	i	PRON
ejpam-4412	182	33	containing	contain	VERB
ejpam-4412	182	34	0	0	NUM
ejpam-4412	182	35	with	with	ADP
ejpam-4412	182	36	f(0	f(0	NOUN
ejpam-4412	182	37	)	)	PUNCT
ejpam-4412	182	38	≤	≤	NOUN
ejpam-4412	182	39	0	0	PUNCT
ejpam-4412	183	1	if	if	SCONJ
ejpam-4412	183	2	and	and	CCONJ
ejpam-4412	183	3	only	only	ADV
ejpam-4412	183	4	if	if	SCONJ
ejpam-4412	183	5	f(a∗xa	f(a∗xa	NOUN
ejpam-4412	183	6	)	)	PUNCT
ejpam-4412	183	7	≤	≤	NOUN
ejpam-4412	183	8	a∗f(x)a	a∗f(x)a	PROPN
ejpam-4412	183	9	for	for	ADP
ejpam-4412	183	10	every	every	DET
ejpam-4412	183	11	self	self	NOUN
ejpam-4412	183	12	-	-	PUNCT
ejpam-4412	183	13	adjoint	adjoint	NOUN
ejpam-4412	183	14	x	x	PUNCT
ejpam-4412	183	15	with	with	ADP
ejpam-4412	183	16	spectrum	spectrum	NOUN
ejpam-4412	183	17	in	in	ADP
ejpam-4412	183	18	i	i	PRON
ejpam-4412	183	19	and	and	CCONJ
ejpam-4412	183	20	every	every	DET
ejpam-4412	183	21	contraction	contraction	NOUN
ejpam-4412	183	22	a.	a.	NOUN
ejpam-4412	183	23	theorem	theorem	NOUN
ejpam-4412	183	24	5	5	NUM
ejpam-4412	183	25	.	.	PUNCT
ejpam-4412	183	26	(	(	PUNCT
ejpam-4412	183	27	[	[	X
ejpam-4412	183	28	11	11	NUM
ejpam-4412	183	29	]	]	PUNCT
ejpam-4412	183	30	)	)	PUNCT
ejpam-4412	183	31	let	let	VERB
ejpam-4412	183	32	a	a	PRON
ejpam-4412	183	33	and	and	CCONJ
ejpam-4412	183	34	b	b	NOUN
ejpam-4412	183	35	be	be	AUX
ejpam-4412	183	36	bounded	bound	VERB
ejpam-4412	183	37	linear	linear	PROPN
ejpam-4412	183	38	operators	operator	NOUN
ejpam-4412	183	39	on	on	ADP
ejpam-4412	183	40	a	a	DET
ejpam-4412	183	41	hilbert	hilbert	NOUN
ejpam-4412	183	42	space	space	NOUN
ejpam-4412	183	43	h.	h.	PROPN
ejpam-4412	183	44	then	then	ADV
ejpam-4412	183	45	the	the	DET
ejpam-4412	183	46	following	follow	VERB
ejpam-4412	183	47	are	be	AUX
ejpam-4412	183	48	equivalent	equivalent	ADJ
ejpam-4412	183	49	:	:	PUNCT
ejpam-4412	183	50	(	(	PUNCT
ejpam-4412	183	51	i	i	NOUN
ejpam-4412	183	52	)	)	PUNCT
ejpam-4412	183	53	ran(a	ran(a	PROPN
ejpam-4412	183	54	)	)	PUNCT
ejpam-4412	183	55	⊆	⊆	NUM
ejpam-4412	183	56	ran(b	ran(b	NOUN
ejpam-4412	183	57	)	)	PUNCT
ejpam-4412	183	58	;	;	PUNCT
ejpam-4412	183	59	(	(	PUNCT
ejpam-4412	183	60	ii	ii	NOUN
ejpam-4412	183	61	)	)	PUNCT
ejpam-4412	183	62	aa∗	aa∗	NOUN
ejpam-4412	183	63	≤	≤	NUM
ejpam-4412	183	64	λ2bb∗	λ2bb∗	ADV
ejpam-4412	183	65	for	for	ADP
ejpam-4412	183	66	some	some	DET
ejpam-4412	183	67	λ	λ	NOUN
ejpam-4412	183	68	≥	≥	NOUN
ejpam-4412	183	69	0	0	NUM
ejpam-4412	183	70	;	;	PUNCT
ejpam-4412	183	71	and	and	CCONJ
ejpam-4412	183	72	(	(	PUNCT
ejpam-4412	183	73	i	i	NOUN
ejpam-4412	183	74	)	)	PUNCT
ejpam-4412	183	75	there	there	PRON
ejpam-4412	183	76	exists	exist	VERB
ejpam-4412	183	77	a	a	DET
ejpam-4412	183	78	bounded	bounded	ADJ
ejpam-4412	183	79	linear	linear	ADJ
ejpam-4412	183	80	operator	operator	NOUN
ejpam-4412	183	81	c	c	PROPN
ejpam-4412	183	82	on	on	ADP
ejpam-4412	183	83	h	h	NOUN
ejpam-4412	183	84	so	so	SCONJ
ejpam-4412	183	85	that	that	SCONJ
ejpam-4412	183	86	a	a	DET
ejpam-4412	183	87	=	=	SYM
ejpam-4412	183	88	bc	bc	PROPN
ejpam-4412	183	89	.	.	PUNCT
ejpam-4412	183	90	lemma	lemma	PROPN
ejpam-4412	183	91	2	2	X
ejpam-4412	183	92	.	.	PUNCT
ejpam-4412	183	93	let	let	VERB
ejpam-4412	183	94	a	a	DET
ejpam-4412	183	95	,	,	PUNCT
ejpam-4412	183	96	b	b	NOUN
ejpam-4412	183	97	and	and	CCONJ
ejpam-4412	183	98	c	c	PROPN
ejpam-4412	183	99	be	be	AUX
ejpam-4412	183	100	positive	positive	ADJ
ejpam-4412	183	101	operators	operator	NOUN
ejpam-4412	183	102	.	.	PUNCT
ejpam-4412	184	1	then	then	ADV
ejpam-4412	184	2	the	the	DET
ejpam-4412	184	3	following	follow	VERB
ejpam-4412	184	4	assertions	assertion	NOUN
ejpam-4412	184	5	hold	hold	VERB
ejpam-4412	184	6	for	for	ADP
ejpam-4412	184	7	each	each	DET
ejpam-4412	184	8	p	p	NOUN
ejpam-4412	184	9	≥	≥	NOUN
ejpam-4412	184	10	0	0	NUM
ejpam-4412	184	11	,	,	PUNCT
ejpam-4412	184	12	r	r	NOUN
ejpam-4412	184	13	∈	∈	PROPN
ejpam-4412	185	1	[	[	X
ejpam-4412	185	2	0	0	NUM
ejpam-4412	185	3	,	,	PUNCT
ejpam-4412	185	4	1	1	NUM
ejpam-4412	185	5	]	]	PUNCT
ejpam-4412	185	6	and	and	CCONJ
ejpam-4412	185	7	0	0	NUM
ejpam-4412	185	8	<	<	X
ejpam-4412	185	9	q	q	X
ejpam-4412	185	10	≤	≤	NUM
ejpam-4412	185	11	1	1	NUM
ejpam-4412	185	12	:	:	PUNCT
ejpam-4412	185	13	(	(	PUNCT
ejpam-4412	185	14	i	i	NOUN
ejpam-4412	185	15	)	)	PUNCT
ejpam-4412	185	16	if	if	SCONJ
ejpam-4412	185	17	(	(	PUNCT
ejpam-4412	185	18	br/2apbr/2	br/2apbr/2	NOUN
ejpam-4412	185	19	)	)	PUNCT
ejpam-4412	185	20	rq	rq	VERB
ejpam-4412	185	21	p+r	p+r	NUM
ejpam-4412	185	22	≥	≥	X
ejpam-4412	185	23	brq	brq	NOUN
ejpam-4412	185	24	and	and	CCONJ
ejpam-4412	185	25	b	b	NOUN
ejpam-4412	185	26	≥	≥	NOUN
ejpam-4412	185	27	c	c	NOUN
ejpam-4412	185	28	,	,	PUNCT
ejpam-4412	185	29	then	then	ADV
ejpam-4412	185	30	(	(	PUNCT
ejpam-4412	185	31	cr/2apcr/2	cr/2apcr/2	NOUN
ejpam-4412	185	32	)	)	PUNCT
ejpam-4412	185	33	rq	rq	VERB
ejpam-4412	185	34	p+r	p+r	NUM
ejpam-4412	185	35	≥	≥	NOUN
ejpam-4412	185	36	crq	crq	NOUN
ejpam-4412	185	37	.	.	PUNCT
ejpam-4412	186	1	(	(	PUNCT
ejpam-4412	186	2	ii	ii	NOUN
ejpam-4412	186	3	)	)	PUNCT
ejpam-4412	186	4	if	if	SCONJ
ejpam-4412	186	5	a	a	DET
ejpam-4412	186	6	≥	≥	NOUN
ejpam-4412	186	7	b	b	NOUN
ejpam-4412	186	8	,	,	PUNCT
ejpam-4412	186	9	brq	brq	ADJ
ejpam-4412	186	10	≥	≥	NOUN
ejpam-4412	186	11	(	(	PUNCT
ejpam-4412	186	12	br/2cpbr/2	br/2cpbr/2	PROPN
ejpam-4412	186	13	)	)	PUNCT
ejpam-4412	186	14	rq	rq	VERB
ejpam-4412	186	15	p+r	p+r	NOUN
ejpam-4412	186	16	and	and	CCONJ
ejpam-4412	186	17	the	the	DET
ejpam-4412	186	18	condition	condition	NOUN
ejpam-4412	186	19	if	if	SCONJ
ejpam-4412	186	20	lim	lim	PROPN
ejpam-4412	186	21	n→∞	n→∞	PRON
ejpam-4412	186	22	b1/2xn	b1/2xn	PROPN
ejpam-4412	186	23	=	=	SYM
ejpam-4412	186	24	0	0	PUNCT
ejpam-4412	186	25	and	and	CCONJ
ejpam-4412	186	26	lim	lim	PROPN
ejpam-4412	186	27	n→∞	n→∞	X
ejpam-4412	186	28	a1/2xn	a1/2xn	PROPN
ejpam-4412	186	29	exists	exist	VERB
ejpam-4412	186	30	,	,	PUNCT
ejpam-4412	186	31	then	then	ADV
ejpam-4412	186	32	lim	lim	PROPN
ejpam-4412	186	33	n→∞	n→∞	X
ejpam-4412	186	34	a1/2xn	a1/2xn	PROPN
ejpam-4412	186	35	=	=	PUNCT
ejpam-4412	186	36	0	0	NUM
ejpam-4412	186	37	for	for	ADP
ejpam-4412	186	38	any	any	DET
ejpam-4412	186	39	sequence	sequence	NOUN
ejpam-4412	186	40	of	of	ADP
ejpam-4412	186	41	vectors	vector	NOUN
ejpam-4412	186	42	{	{	PUNCT
ejpam-4412	186	43	xn	xn	NUM
ejpam-4412	186	44	}	}	PUNCT
ejpam-4412	186	45	(	(	PUNCT
ejpam-4412	186	46	8)	8)	NUM
ejpam-4412	186	47	hold	hold	NOUN
ejpam-4412	186	48	,	,	PUNCT
ejpam-4412	186	49	then	then	ADV
ejpam-4412	186	50	arq	arq	VERB
ejpam-4412	186	51	≥	≥	PRON
ejpam-4412	186	52	(	(	PUNCT
ejpam-4412	186	53	ar/2cpar/2	ar/2cpar/2	NOUN
ejpam-4412	186	54	)	)	PUNCT
ejpam-4412	186	55	rq	rq	VERB
ejpam-4412	186	56	p+r	p+r	NUM
ejpam-4412	186	57	.	.	PUNCT
ejpam-4412	187	1	lemma	lemma	PROPN
ejpam-4412	187	2	2	2	PROPN
ejpam-4412	187	3	can	can	AUX
ejpam-4412	187	4	be	be	AUX
ejpam-4412	187	5	obtained	obtain	VERB
ejpam-4412	187	6	as	as	ADP
ejpam-4412	187	7	an	an	DET
ejpam-4412	187	8	application	application	NOUN
ejpam-4412	187	9	of	of	ADP
ejpam-4412	187	10	the	the	DET
ejpam-4412	187	11	following	follow	VERB
ejpam-4412	187	12	results	result	NOUN
ejpam-4412	187	13	.	.	PUNCT
ejpam-4412	188	1	theorem	theorem	NOUN
ejpam-4412	188	2	6	6	NUM
ejpam-4412	188	3	.	.	PUNCT
ejpam-4412	189	1	(	(	PUNCT
ejpam-4412	189	2	[	[	X
ejpam-4412	189	3	11	11	NUM
ejpam-4412	189	4	]	]	PUNCT
ejpam-4412	189	5	)	)	PUNCT
ejpam-4412	189	6	let	let	VERB
ejpam-4412	189	7	a	a	PRON
ejpam-4412	189	8	and	and	CCONJ
ejpam-4412	189	9	b	b	NOUN
ejpam-4412	189	10	be	be	AUX
ejpam-4412	189	11	bounded	bound	VERB
ejpam-4412	189	12	linear	linear	PROPN
ejpam-4412	189	13	operators	operator	NOUN
ejpam-4412	189	14	on	on	ADP
ejpam-4412	189	15	a	a	DET
ejpam-4412	189	16	hilbert	hilbert	NOUN
ejpam-4412	189	17	space	space	NOUN
ejpam-4412	189	18	h.	h.	PROPN
ejpam-4412	189	19	then	then	ADV
ejpam-4412	189	20	the	the	DET
ejpam-4412	189	21	following	follow	VERB
ejpam-4412	189	22	are	be	AUX
ejpam-4412	189	23	equivalent	equivalent	ADJ
ejpam-4412	189	24	:	:	PUNCT
ejpam-4412	189	25	(	(	PUNCT
ejpam-4412	189	26	i	i	NOUN
ejpam-4412	189	27	)	)	PUNCT
ejpam-4412	190	1	ran(a	ran(a	PROPN
ejpam-4412	190	2	)	)	PUNCT
ejpam-4412	191	1	⊆	⊆	NUM
ejpam-4412	191	2	ran(b	ran(b	NOUN
ejpam-4412	191	3	)	)	PUNCT
ejpam-4412	191	4	;	;	PUNCT
ejpam-4412	191	5	(	(	PUNCT
ejpam-4412	191	6	ii	ii	NOUN
ejpam-4412	191	7	)	)	PUNCT
ejpam-4412	191	8	aa∗	aa∗	NOUN
ejpam-4412	191	9	≤	≤	NUM
ejpam-4412	191	10	λ2bb∗	λ2bb∗	ADV
ejpam-4412	191	11	for	for	ADP
ejpam-4412	191	12	some	some	DET
ejpam-4412	191	13	λ	λ	NOUN
ejpam-4412	191	14	≥	≥	NOUN
ejpam-4412	191	15	0	0	NUM
ejpam-4412	191	16	;	;	PUNCT
ejpam-4412	191	17	and	and	CCONJ
ejpam-4412	191	18	(	(	PUNCT
ejpam-4412	191	19	iii	iii	X
ejpam-4412	191	20	)	)	PUNCT
ejpam-4412	191	21	there	there	PRON
ejpam-4412	191	22	exists	exist	VERB
ejpam-4412	191	23	a	a	DET
ejpam-4412	191	24	bounded	bounded	ADJ
ejpam-4412	191	25	linear	linear	ADJ
ejpam-4412	191	26	operator	operator	NOUN
ejpam-4412	191	27	c	c	PROPN
ejpam-4412	191	28	on	on	ADP
ejpam-4412	191	29	h	h	NOUN
ejpam-4412	191	30	so	so	SCONJ
ejpam-4412	191	31	that	that	SCONJ
ejpam-4412	191	32	a	a	DET
ejpam-4412	191	33	=	=	SYM
ejpam-4412	191	34	bc	bc	PROPN
ejpam-4412	191	35	.	.	PUNCT
ejpam-4412	192	1	moreover	moreover	ADV
ejpam-4412	192	2	,	,	PUNCT
ejpam-4412	192	3	if	if	SCONJ
ejpam-4412	192	4	(	(	PUNCT
ejpam-4412	192	5	i	i	NOUN
ejpam-4412	192	6	)	)	PUNCT
ejpam-4412	192	7	,	,	PUNCT
ejpam-4412	192	8	(	(	PUNCT
ejpam-4412	192	9	ii	ii	NOUN
ejpam-4412	192	10	)	)	PUNCT
ejpam-4412	192	11	and	and	CCONJ
ejpam-4412	192	12	(	(	PUNCT
ejpam-4412	192	13	iii	iii	X
ejpam-4412	192	14	)	)	PUNCT
ejpam-4412	192	15	are	be	AUX
ejpam-4412	192	16	valid	valid	ADJ
ejpam-4412	192	17	,	,	PUNCT
ejpam-4412	192	18	then	then	ADV
ejpam-4412	192	19	there	there	PRON
ejpam-4412	192	20	exists	exist	VERB
ejpam-4412	192	21	a	a	DET
ejpam-4412	192	22	unique	unique	ADJ
ejpam-4412	192	23	operator	operator	NOUN
ejpam-4412	192	24	c	c	NOUN
ejpam-4412	192	25	so	so	SCONJ
ejpam-4412	192	26	that	that	SCONJ
ejpam-4412	192	27	(	(	PUNCT
ejpam-4412	192	28	a	a	X
ejpam-4412	192	29	)	)	PUNCT
ejpam-4412	192	30	∥c∥2	∥c∥2	NOUN
ejpam-4412	192	31	=	=	PUNCT
ejpam-4412	192	32	inf{µ	inf{µ	NOUN
ejpam-4412	192	33	:	:	PUNCT
ejpam-4412	192	34	aa∗	aa∗	NOUN
ejpam-4412	192	35	≤	≤	PROPN
ejpam-4412	192	36	µbb∗	µbb∗	PROPN
ejpam-4412	192	37	}	}	PUNCT
ejpam-4412	192	38	;	;	PUNCT
ejpam-4412	192	39	(	(	PUNCT
ejpam-4412	192	40	b	b	X
ejpam-4412	192	41	)	)	PUNCT
ejpam-4412	192	42	ker(a	ker(a	NOUN
ejpam-4412	192	43	)	)	PUNCT
ejpam-4412	193	1	=	=	PUNCT
ejpam-4412	193	2	ker(c	ker(c	PROPN
ejpam-4412	193	3	)	)	PUNCT
ejpam-4412	193	4	;	;	PUNCT
ejpam-4412	193	5	and	and	CCONJ
ejpam-4412	193	6	(	(	PUNCT
ejpam-4412	193	7	c	c	X
ejpam-4412	193	8	)	)	PUNCT
ejpam-4412	193	9	ran(c	ran(c	NOUN
ejpam-4412	193	10	)	)	PUNCT
ejpam-4412	193	11	⊆	⊆	NUM
ejpam-4412	193	12	ran(b∗	ran(b∗	X
ejpam-4412	193	13	)	)	PUNCT
ejpam-4412	193	14	.	.	PUNCT
ejpam-4412	194	1	theorem	theorem	VERB
ejpam-4412	194	2	7	7	NUM
ejpam-4412	194	3	.	.	PUNCT
ejpam-4412	195	1	(	(	PUNCT
ejpam-4412	195	2	[	[	X
ejpam-4412	195	3	16	16	NUM
ejpam-4412	195	4	]	]	PUNCT
ejpam-4412	195	5	)	)	PUNCT
ejpam-4412	195	6	let	let	VERB
ejpam-4412	195	7	x	x	PUNCT
ejpam-4412	195	8	and	and	CCONJ
ejpam-4412	195	9	a	a	DET
ejpam-4412	195	10	be	be	AUX
ejpam-4412	195	11	bounded	bound	VERB
ejpam-4412	195	12	linear	linear	ADJ
ejpam-4412	195	13	operator	operator	NOUN
ejpam-4412	195	14	on	on	ADP
ejpam-4412	195	15	a	a	DET
ejpam-4412	195	16	hilbert	hilbert	NOUN
ejpam-4412	195	17	space	space	NOUN
ejpam-4412	195	18	h.	h.	PROPN
ejpam-4412	195	19	we	we	PRON
ejpam-4412	195	20	suppose	suppose	VERB
ejpam-4412	195	21	that	that	SCONJ
ejpam-4412	195	22	a	a	DET
ejpam-4412	195	23	≥	≥	NOUN
ejpam-4412	195	24	0	0	NUM
ejpam-4412	195	25	and	and	CCONJ
ejpam-4412	195	26	∥x∥	∥x∥	NOUN
ejpam-4412	195	27	≤	≤	ADV
ejpam-4412	195	28	1	1	NUM
ejpam-4412	195	29	.	.	PUNCT
ejpam-4412	196	1	if	if	SCONJ
ejpam-4412	196	2	f	f	PROPN
ejpam-4412	196	3	is	be	AUX
ejpam-4412	196	4	an	an	DET
ejpam-4412	196	5	operator	operator	NOUN
ejpam-4412	196	6	monotone	monotone	NOUN
ejpam-4412	196	7	function	function	NOUN
ejpam-4412	196	8	defined	define	VERB
ejpam-4412	196	9	on	on	ADP
ejpam-4412	196	10	[	[	X
ejpam-4412	196	11	0,∞	0,∞	NOUN
ejpam-4412	196	12	)	)	PUNCT
ejpam-4412	196	13	,	,	PUNCT
ejpam-4412	196	14	then	then	ADV
ejpam-4412	196	15	x∗f(a)x	x∗f(a)x	PROPN
ejpam-4412	196	16	≤	≤	PROPN
ejpam-4412	196	17	f(x∗ax	f(x∗ax	PROPN
ejpam-4412	196	18	)	)	PUNCT
ejpam-4412	196	19	.	.	PUNCT
ejpam-4412	197	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	197	2	,	,	PUNCT
ejpam-4412	197	3	n.	n.	NOUN
ejpam-4412	197	4	h.	h.	PROPN
ejpam-4412	197	5	altaweel	altaweel	PROPN
ejpam-4412	197	6	/	/	SYM
ejpam-4412	197	7	eur	eur	PROPN
ejpam-4412	197	8	.	.	PUNCT
ejpam-4412	198	1	j.	j.	PROPN
ejpam-4412	198	2	pure	pure	PROPN
ejpam-4412	198	3	appl	appl	PROPN
ejpam-4412	198	4	.	.	PROPN
ejpam-4412	198	5	math	math	PROPN
ejpam-4412	198	6	,	,	PUNCT
ejpam-4412	198	7	15	15	NUM
ejpam-4412	198	8	(	(	PUNCT
ejpam-4412	198	9	3	3	NUM
ejpam-4412	198	10	)	)	PUNCT
ejpam-4412	198	11	(	(	PUNCT
ejpam-4412	198	12	2022	2022	NUM
ejpam-4412	198	13	)	)	PUNCT
ejpam-4412	198	14	,	,	PUNCT
ejpam-4412	198	15	1067	1067	NUM
ejpam-4412	198	16	-	-	SYM
ejpam-4412	198	17	1089	1089	NUM
ejpam-4412	198	18	1074	1074	NUM
ejpam-4412	198	19	we	we	PRON
ejpam-4412	198	20	remark	remark	VERB
ejpam-4412	198	21	that	that	SCONJ
ejpam-4412	198	22	the	the	DET
ejpam-4412	198	23	condition	condition	NOUN
ejpam-4412	198	24	(	(	PUNCT
ejpam-4412	198	25	c	c	NOUN
ejpam-4412	198	26	)	)	PUNCT
ejpam-4412	198	27	of	of	ADP
ejpam-4412	198	28	theorem	theorem	NOUN
ejpam-4412	198	29	6	6	NUM
ejpam-4412	198	30	is	be	AUX
ejpam-4412	198	31	equivalent	equivalent	ADJ
ejpam-4412	198	32	to	to	ADP
ejpam-4412	198	33	(	(	PUNCT
ejpam-4412	198	34	c′	c′	NUM
ejpam-4412	198	35	):	):	PUNCT
ejpam-4412	198	36	ran(c	ran(c	NOUN
ejpam-4412	198	37	)	)	PUNCT
ejpam-4412	198	38	⊆	⊆	NUM
ejpam-4412	198	39	ran(b∗	ran(b∗	X
ejpam-4412	198	40	)	)	PUNCT
ejpam-4412	198	41	.	.	PUNCT
ejpam-4412	199	1	here	here	ADV
ejpam-4412	199	2	we	we	PRON
ejpam-4412	199	3	consider	consider	VERB
ejpam-4412	199	4	when	when	SCONJ
ejpam-4412	199	5	the	the	DET
ejpam-4412	199	6	equality	equality	NOUN
ejpam-4412	199	7	of	of	ADP
ejpam-4412	199	8	(	(	PUNCT
ejpam-4412	199	9	c′	c′	NOUN
ejpam-4412	199	10	)	)	PUNCT
ejpam-4412	199	11	holds	hold	VERB
ejpam-4412	199	12	.	.	PUNCT
ejpam-4412	200	1	lemma	lemma	PROPN
ejpam-4412	200	2	3	3	NUM
ejpam-4412	200	3	.	.	PUNCT
ejpam-4412	201	1	(	(	PUNCT
ejpam-4412	201	2	[	[	X
ejpam-4412	201	3	33	33	NUM
ejpam-4412	201	4	]	]	PUNCT
ejpam-4412	201	5	)	)	PUNCT
ejpam-4412	201	6	let	let	VERB
ejpam-4412	201	7	a	a	PRON
ejpam-4412	201	8	and	and	CCONJ
ejpam-4412	201	9	b	b	NOUN
ejpam-4412	201	10	be	be	AUX
ejpam-4412	201	11	operators	operator	NOUN
ejpam-4412	201	12	which	which	PRON
ejpam-4412	201	13	satisfy	satisfy	VERB
ejpam-4412	201	14	(	(	PUNCT
ejpam-4412	201	15	i	i	NOUN
ejpam-4412	201	16	)	)	PUNCT
ejpam-4412	201	17	,	,	PUNCT
ejpam-4412	201	18	(	(	PUNCT
ejpam-4412	201	19	ii	ii	NOUN
ejpam-4412	201	20	)	)	PUNCT
ejpam-4412	201	21	and	and	CCONJ
ejpam-4412	201	22	(	(	PUNCT
ejpam-4412	201	23	iii	iii	NOUN
ejpam-4412	201	24	)	)	PUNCT
ejpam-4412	201	25	of	of	ADP
ejpam-4412	201	26	theorem	theorem	NOUN
ejpam-4412	201	27	6	6	NUM
ejpam-4412	201	28	and	and	CCONJ
ejpam-4412	201	29	c	c	PROPN
ejpam-4412	201	30	be	be	AUX
ejpam-4412	201	31	the	the	DET
ejpam-4412	201	32	operator	operator	NOUN
ejpam-4412	201	33	which	which	PRON
ejpam-4412	201	34	is	be	AUX
ejpam-4412	201	35	given	give	VERB
ejpam-4412	201	36	in	in	ADP
ejpam-4412	201	37	(	(	PUNCT
ejpam-4412	201	38	iii	iii	NOUN
ejpam-4412	201	39	)	)	PUNCT
ejpam-4412	201	40	and	and	CCONJ
ejpam-4412	201	41	determined	determine	VERB
ejpam-4412	201	42	uniquely	uniquely	ADV
ejpam-4412	201	43	by	by	ADP
ejpam-4412	201	44	(	(	PUNCT
ejpam-4412	201	45	a	a	NOUN
ejpam-4412	201	46	)	)	PUNCT
ejpam-4412	201	47	,	,	PUNCT
ejpam-4412	201	48	(	(	PUNCT
ejpam-4412	201	49	b	b	X
ejpam-4412	201	50	)	)	PUNCT
ejpam-4412	201	51	and	and	CCONJ
ejpam-4412	201	52	(	(	PUNCT
ejpam-4412	201	53	c	c	NOUN
ejpam-4412	201	54	)	)	PUNCT
ejpam-4412	201	55	of	of	ADP
ejpam-4412	201	56	theorem	theorem	NOUN
ejpam-4412	201	57	6	6	NUM
ejpam-4412	201	58	.	.	PUNCT
ejpam-4412	202	1	then	then	ADV
ejpam-4412	202	2	the	the	DET
ejpam-4412	202	3	following	follow	VERB
ejpam-4412	202	4	assertions	assertion	NOUN
ejpam-4412	202	5	are	be	AUX
ejpam-4412	202	6	mutually	mutually	ADV
ejpam-4412	202	7	equivalent	equivalent	ADJ
ejpam-4412	202	8	:	:	PUNCT
ejpam-4412	202	9	(	(	PUNCT
ejpam-4412	202	10	i	i	NOUN
ejpam-4412	202	11	)	)	PUNCT
ejpam-4412	202	12	ran(c	ran(c	NOUN
ejpam-4412	202	13	)	)	PUNCT
ejpam-4412	202	14	=	=	SYM
ejpam-4412	202	15	ran(b∗	ran(b∗	PROPN
ejpam-4412	202	16	)	)	PUNCT
ejpam-4412	202	17	.	.	PUNCT
ejpam-4412	203	1	(	(	PUNCT
ejpam-4412	203	2	ii	ii	NOUN
ejpam-4412	203	3	)	)	PUNCT
ejpam-4412	203	4	if	if	SCONJ
ejpam-4412	203	5	lim	lim	PROPN
ejpam-4412	203	6	n→∞	n→∞	PRON
ejpam-4412	203	7	a∗xn	a∗xn	PROPN
ejpam-4412	203	8	=	=	PUNCT
ejpam-4412	203	9	0	0	PUNCT
ejpam-4412	203	10	and	and	CCONJ
ejpam-4412	203	11	lim	lim	PROPN
ejpam-4412	203	12	n→∞	n→∞	NUM
ejpam-4412	203	13	b∗xn	b∗xn	NOUN
ejpam-4412	203	14	exists	exist	VERB
ejpam-4412	203	15	,	,	PUNCT
ejpam-4412	203	16	then	then	ADV
ejpam-4412	203	17	lim	lim	PROPN
ejpam-4412	203	18	n→∞	n→∞	PRON
ejpam-4412	203	19	b∗xn	b∗xn	VERB
ejpam-4412	203	20	=	=	NOUN
ejpam-4412	203	21	0	0	NUM
ejpam-4412	204	1	for	for	ADP
ejpam-4412	204	2	any	any	DET
ejpam-4412	204	3	sequence	sequence	NOUN
ejpam-4412	204	4	of	of	ADP
ejpam-4412	204	5	vectors	vector	NOUN
ejpam-4412	204	6	{	{	PUNCT
ejpam-4412	204	7	xn	xn	NUM
ejpam-4412	204	8	}	}	PUNCT
ejpam-4412	204	9	.	.	PUNCT
ejpam-4412	205	1	we	we	PRON
ejpam-4412	205	2	also	also	ADV
ejpam-4412	205	3	prepare	prepare	VERB
ejpam-4412	205	4	the	the	DET
ejpam-4412	205	5	following	follow	VERB
ejpam-4412	205	6	lemma	lemma	PROPN
ejpam-4412	205	7	in	in	ADP
ejpam-4412	205	8	order	order	NOUN
ejpam-4412	205	9	to	to	PART
ejpam-4412	205	10	give	give	VERB
ejpam-4412	205	11	a	a	DET
ejpam-4412	205	12	proof	proof	NOUN
ejpam-4412	205	13	of	of	ADP
ejpam-4412	205	14	lemma	lemma	PROPN
ejpam-4412	205	15	2	2	NUM
ejpam-4412	205	16	.	.	PUNCT
ejpam-4412	206	1	lemma	lemma	PROPN
ejpam-4412	206	2	4	4	NUM
ejpam-4412	206	3	.	.	PUNCT
ejpam-4412	207	1	(	(	PUNCT
ejpam-4412	207	2	[	[	X
ejpam-4412	207	3	33	33	NUM
ejpam-4412	207	4	]	]	PUNCT
ejpam-4412	207	5	)	)	PUNCT
ejpam-4412	207	6	let	let	VERB
ejpam-4412	207	7	s	s	PRON
ejpam-4412	207	8	be	be	AUX
ejpam-4412	207	9	a	a	DET
ejpam-4412	207	10	positive	positive	ADJ
ejpam-4412	207	11	operator	operator	NOUN
ejpam-4412	207	12	and	and	CCONJ
ejpam-4412	208	1	0	0	NUM
ejpam-4412	208	2	<	<	X
ejpam-4412	208	3	q	q	X
ejpam-4412	208	4	≤	≤	NUM
ejpam-4412	208	5	1	1	NUM
ejpam-4412	208	6	.	.	PUNCT
ejpam-4412	209	1	if	if	SCONJ
ejpam-4412	209	2	lim	lim	PROPN
ejpam-4412	209	3	n→∞	n→∞	X
ejpam-4412	209	4	sxn	sxn	NOUN
ejpam-4412	209	5	=	=	SYM
ejpam-4412	209	6	0	0	PROPN
ejpam-4412	209	7	and	and	CCONJ
ejpam-4412	209	8	lim	lim	PROPN
ejpam-4412	209	9	n→∞	n→∞	PROPN
ejpam-4412	209	10	sqxn	sqxn	PROPN
ejpam-4412	209	11	exists	exist	VERB
ejpam-4412	209	12	,	,	PUNCT
ejpam-4412	209	13	then	then	ADV
ejpam-4412	209	14	lim	lim	PROPN
ejpam-4412	209	15	n→∞	n→∞	X
ejpam-4412	210	1	sqxn	sqxn	PROPN
ejpam-4412	210	2	=	=	NOUN
ejpam-4412	210	3	0	0	NUM
ejpam-4412	210	4	for	for	ADP
ejpam-4412	210	5	any	any	DET
ejpam-4412	210	6	sequence	sequence	NOUN
ejpam-4412	210	7	of	of	ADP
ejpam-4412	210	8	vectors	vector	NOUN
ejpam-4412	210	9	{	{	PUNCT
ejpam-4412	210	10	xn	xn	NUM
ejpam-4412	210	11	}	}	PUNCT
ejpam-4412	210	12	.	.	PUNCT
ejpam-4412	211	1	proof	proof	NOUN
ejpam-4412	211	2	.	.	PUNCT
ejpam-4412	212	1	[	[	X
ejpam-4412	212	2	proof	proof	NOUN
ejpam-4412	212	3	of	of	ADP
ejpam-4412	212	4	lemma	lemma	PROPN
ejpam-4412	212	5	2	2	NUM
ejpam-4412	212	6	]	]	PUNCT
ejpam-4412	212	7	(	(	PUNCT
ejpam-4412	212	8	i	i	NOUN
ejpam-4412	212	9	)	)	PUNCT
ejpam-4412	212	10	the	the	DET
ejpam-4412	212	11	hypothesis	hypothesis	NOUN
ejpam-4412	212	12	b	b	PROPN
ejpam-4412	212	13	≥	≥	PROPN
ejpam-4412	212	14	c	c	PROPN
ejpam-4412	212	15	ensures	ensure	VERB
ejpam-4412	212	16	then	then	ADV
ejpam-4412	212	17	bt	bt	VERB
ejpam-4412	212	18	≥	≥	PROPN
ejpam-4412	212	19	ct	ct	NUM
ejpam-4412	212	20	for	for	ADP
ejpam-4412	212	21	each	each	DET
ejpam-4412	212	22	t	t	NOUN
ejpam-4412	212	23	∈	∈	PROPN
ejpam-4412	212	24	(	(	PUNCT
ejpam-4412	212	25	0	0	NUM
ejpam-4412	212	26	,	,	PUNCT
ejpam-4412	212	27	1	1	NUM
ejpam-4412	212	28	]	]	PUNCT
ejpam-4412	212	29	by	by	ADP
ejpam-4412	212	30	löwner	löwner	NOUN
ejpam-4412	212	31	-	-	PUNCT
ejpam-4412	212	32	heinz	heinz	NOUN
ejpam-4412	212	33	theorem	theorem	NOUN
ejpam-4412	212	34	.	.	PUNCT
ejpam-4412	213	1	by	by	ADP
ejpam-4412	213	2	theorem	theorem	NOUN
ejpam-4412	213	3	6	6	NUM
ejpam-4412	213	4	,	,	PUNCT
ejpam-4412	213	5	there	there	PRON
ejpam-4412	213	6	exists	exist	VERB
ejpam-4412	213	7	an	an	DET
ejpam-4412	213	8	operator	operator	NOUN
ejpam-4412	213	9	x	x	PUNCT
ejpam-4412	213	10	with	with	ADP
ejpam-4412	213	11	∥x∥	∥x∥	NOUN
ejpam-4412	213	12	≤	≤	NOUN
ejpam-4412	213	13	1	1	NUM
ejpam-4412	213	14	such	such	ADJ
ejpam-4412	213	15	that	that	DET
ejpam-4412	213	16	b	b	NOUN
ejpam-4412	213	17	t	t	NOUN
ejpam-4412	213	18	2x	2x	NUM
ejpam-4412	213	19	=	=	SYM
ejpam-4412	213	20	x∗b	x∗b	PROPN
ejpam-4412	213	21	t	t	PROPN
ejpam-4412	213	22	2	2	NUM
ejpam-4412	213	23	=	=	SYM
ejpam-4412	213	24	c	c	PROPN
ejpam-4412	213	25	t	t	PROPN
ejpam-4412	213	26	2	2	NUM
ejpam-4412	213	27	.	.	PUNCT
ejpam-4412	214	1	(	(	PUNCT
ejpam-4412	214	2	9	9	NUM
ejpam-4412	214	3	)	)	PUNCT
ejpam-4412	214	4	then	then	ADV
ejpam-4412	214	5	we	we	PRON
ejpam-4412	214	6	have	have	AUX
ejpam-4412	214	7	(	(	PUNCT
ejpam-4412	214	8	cr/2apcr/2	cr/2apcr/2	NOUN
ejpam-4412	214	9	)	)	PUNCT
ejpam-4412	214	10	rq	rq	VERB
ejpam-4412	214	11	p+r	p+r	NUM
ejpam-4412	214	12	=	=	SYM
ejpam-4412	214	13	(	(	PUNCT
ejpam-4412	214	14	x∗br/2apbr/2x	x∗br/2apbr/2x	PROPN
ejpam-4412	214	15	)	)	PUNCT
ejpam-4412	214	16	rq	rq	VERB
ejpam-4412	214	17	p+r	p+r	NUM
ejpam-4412	214	18	≥	≥	PROPN
ejpam-4412	214	19	x∗(br/2apbr/2	x∗(br/2apbr/2	PROPN
ejpam-4412	214	20	)	)	PUNCT
ejpam-4412	215	1	rq	rq	VERB
ejpam-4412	215	2	p+rx	p+rx	PROPN
ejpam-4412	215	3	(	(	PUNCT
ejpam-4412	215	4	by	by	ADP
ejpam-4412	215	5	theorem	theorem	NOUN
ejpam-4412	215	6	7	7	NUM
ejpam-4412	215	7	)	)	PUNCT
ejpam-4412	215	8	≥	≥	NOUN
ejpam-4412	215	9	x∗brqx	x∗brqx	PROPN
ejpam-4412	215	10	(	(	PUNCT
ejpam-4412	215	11	by	by	ADP
ejpam-4412	215	12	the	the	DET
ejpam-4412	215	13	hypothesis	hypothesis	NOUN
ejpam-4412	215	14	)	)	PUNCT
ejpam-4412	215	15	=	=	PUNCT
ejpam-4412	216	1	x∗(br)qx	x∗(br)qx	X
ejpam-4412	216	2	≥	≥	NUM
ejpam-4412	216	3	(	(	PUNCT
ejpam-4412	216	4	x∗b	x∗b	NOUN
ejpam-4412	216	5	r	r	NOUN
ejpam-4412	216	6	2b	2b	NUM
ejpam-4412	216	7	r	r	NOUN
ejpam-4412	216	8	2x)q	2x)q	NUM
ejpam-4412	216	9	(	(	PUNCT
ejpam-4412	216	10	by	by	ADP
ejpam-4412	216	11	theorem	theorem	NOUN
ejpam-4412	216	12	4	4	NUM
ejpam-4412	216	13	)	)	PUNCT
ejpam-4412	216	14	=	=	PUNCT
ejpam-4412	216	15	(	(	PUNCT
ejpam-4412	216	16	c	c	NOUN
ejpam-4412	216	17	r	r	NOUN
ejpam-4412	216	18	2c	2c	NUM
ejpam-4412	216	19	r	r	NOUN
ejpam-4412	216	20	2	2	NUM
ejpam-4412	216	21	)	)	PUNCT
ejpam-4412	216	22	q	q	NOUN
ejpam-4412	217	1	=	=	NOUN
ejpam-4412	217	2	crq	crq	NOUN
ejpam-4412	217	3	(	(	PUNCT
ejpam-4412	217	4	by	by	ADP
ejpam-4412	217	5	equation	equation	NOUN
ejpam-4412	217	6	(	(	PUNCT
ejpam-4412	217	7	9	9	NUM
ejpam-4412	217	8	)	)	PUNCT
ejpam-4412	217	9	)	)	PUNCT
ejpam-4412	217	10	.	.	PUNCT
ejpam-4412	218	1	(	(	PUNCT
ejpam-4412	218	2	ii	ii	X
ejpam-4412	218	3	)	)	PUNCT
ejpam-4412	218	4	the	the	DET
ejpam-4412	218	5	hypothesis	hypothesis	NOUN
ejpam-4412	218	6	a	a	DET
ejpam-4412	218	7	≥	≥	NOUN
ejpam-4412	218	8	b	b	NOUN
ejpam-4412	218	9	ensures	ensure	NOUN
ejpam-4412	218	10	as	as	ADP
ejpam-4412	218	11	≥	≥	NOUN
ejpam-4412	218	12	bs	bs	PROPN
ejpam-4412	218	13	for	for	ADP
ejpam-4412	218	14	s	s	PROPN
ejpam-4412	218	15	∈	∈	PROPN
ejpam-4412	218	16	(	(	PUNCT
ejpam-4412	218	17	0	0	NUM
ejpam-4412	218	18	,	,	PUNCT
ejpam-4412	218	19	1	1	NUM
ejpam-4412	218	20	]	]	PUNCT
ejpam-4412	218	21	by	by	ADP
ejpam-4412	218	22	löwner	löwner	NOUN
ejpam-4412	218	23	-	-	PUNCT
ejpam-4412	218	24	heinz	heinz	NOUN
ejpam-4412	218	25	theorem	theorem	NOUN
ejpam-4412	218	26	.	.	PUNCT
ejpam-4412	219	1	by	by	ADP
ejpam-4412	219	2	theorem	theorem	NOUN
ejpam-4412	219	3	6	6	NUM
ejpam-4412	219	4	,	,	PUNCT
ejpam-4412	219	5	there	there	PRON
ejpam-4412	219	6	exists	exist	VERB
ejpam-4412	219	7	an	an	DET
ejpam-4412	219	8	operator	operator	NOUN
ejpam-4412	219	9	x	x	PUNCT
ejpam-4412	219	10	with	with	ADP
ejpam-4412	219	11	∥x∥	∥x∥	NOUN
ejpam-4412	219	12	≤	≤	NOUN
ejpam-4412	219	13	1	1	NUM
ejpam-4412	219	14	such	such	ADJ
ejpam-4412	219	15	that	that	SCONJ
ejpam-4412	219	16	as/2x	as/2x	NOUN
ejpam-4412	219	17	=	=	SYM
ejpam-4412	219	18	x∗as/2	x∗as/2	NOUN
ejpam-4412	219	19	=	=	SYM
ejpam-4412	219	20	bs/2	bs/2	X
ejpam-4412	219	21	.	.	PUNCT
ejpam-4412	220	1	(	(	PUNCT
ejpam-4412	220	2	10	10	NUM
ejpam-4412	220	3	)	)	PUNCT
ejpam-4412	220	4	then	then	ADV
ejpam-4412	220	5	we	we	PRON
ejpam-4412	220	6	have	have	AUX
ejpam-4412	220	7	x∗(ar/2cpar/2	x∗(ar/2cpar/2	PROPN
ejpam-4412	220	8	)	)	PUNCT
ejpam-4412	220	9	rq	rq	VERB
ejpam-4412	220	10	p+rx	p+rx	NOUN
ejpam-4412	220	11	≤	≤	NOUN
ejpam-4412	220	12	(	(	PUNCT
ejpam-4412	220	13	x∗ar/2cpar/2x	x∗ar/2cpar/2x	NOUN
ejpam-4412	220	14	)	)	PUNCT
ejpam-4412	220	15	rq	rq	VERB
ejpam-4412	220	16	p+r	p+r	X
ejpam-4412	220	17	(	(	PUNCT
ejpam-4412	220	18	by	by	ADP
ejpam-4412	220	19	theorem	theorem	NOUN
ejpam-4412	220	20	7	7	NUM
ejpam-4412	220	21	)	)	PUNCT
ejpam-4412	220	22	=	=	NOUN
ejpam-4412	220	23	(	(	PUNCT
ejpam-4412	220	24	br/2cpbr/2	br/2cpbr/2	PROPN
ejpam-4412	220	25	)	)	PUNCT
ejpam-4412	220	26	rq	rq	VERB
ejpam-4412	220	27	p+r	p+r	NUM
ejpam-4412	220	28	≤	≤	NUM
ejpam-4412	220	29	brq	brq	NOUN
ejpam-4412	220	30	(	(	PUNCT
ejpam-4412	220	31	by	by	ADP
ejpam-4412	220	32	the	the	DET
ejpam-4412	220	33	hypothesis	hypothesis	NOUN
ejpam-4412	220	34	)	)	PUNCT
ejpam-4412	220	35	=	=	PUNCT
ejpam-4412	221	1	(	(	PUNCT
ejpam-4412	221	2	br)q	br)q	PROPN
ejpam-4412	221	3	=	=	SYM
ejpam-4412	221	4	(	(	PUNCT
ejpam-4412	221	5	x∗a	x∗a	PUNCT
ejpam-4412	221	6	r	r	NOUN
ejpam-4412	221	7	2a	2a	NUM
ejpam-4412	221	8	r	r	NOUN
ejpam-4412	221	9	2x)q	2x)q	NUM
ejpam-4412	221	10	≤	≤	NOUN
ejpam-4412	221	11	x∗arqx	x∗arqx	PROPN
ejpam-4412	221	12	(	(	PUNCT
ejpam-4412	221	13	by	by	ADP
ejpam-4412	221	14	theorem	theorem	NOUN
ejpam-4412	221	15	4	4	NUM
ejpam-4412	221	16	)	)	PUNCT
ejpam-4412	221	17	so	so	SCONJ
ejpam-4412	221	18	that	that	PRON
ejpam-4412	221	19	arq	arq	VERB
ejpam-4412	221	20	≥	≥	PRON
ejpam-4412	221	21	(	(	PUNCT
ejpam-4412	221	22	ar/2cpar/2	ar/2cpar/2	NOUN
ejpam-4412	221	23	)	)	PUNCT
ejpam-4412	221	24	rq	rq	VERB
ejpam-4412	221	25	p+r	p+r	NUM
ejpam-4412	221	26	holds	hold	VERB
ejpam-4412	221	27	on	on	ADP
ejpam-4412	221	28	ran(x	ran(x	NOUN
ejpam-4412	221	29	)	)	PUNCT
ejpam-4412	221	30	.	.	PUNCT
ejpam-4412	222	1	on	on	ADP
ejpam-4412	222	2	the	the	DET
ejpam-4412	222	3	other	other	ADJ
ejpam-4412	222	4	hand	hand	NOUN
ejpam-4412	222	5	,	,	PUNCT
ejpam-4412	222	6	the	the	DET
ejpam-4412	222	7	hypothesis	hypothesis	NOUN
ejpam-4412	222	8	(	(	PUNCT
ejpam-4412	222	9	8)	8)	NUM
ejpam-4412	222	10	implies	imply	VERB
ejpam-4412	222	11	the	the	DET
ejpam-4412	222	12	following	follow	VERB
ejpam-4412	222	13	(	(	PUNCT
ejpam-4412	222	14	11	11	NUM
ejpam-4412	222	15	)	)	PUNCT
ejpam-4412	222	16	if	if	SCONJ
ejpam-4412	222	17	lim	lim	PROPN
ejpam-4412	222	18	n→∞	n→∞	X
ejpam-4412	222	19	br/2xn	br/2xn	PROPN
ejpam-4412	222	20	=	=	SYM
ejpam-4412	222	21	0	0	PUNCT
ejpam-4412	222	22	and	and	CCONJ
ejpam-4412	222	23	lim	lim	PROPN
ejpam-4412	222	24	n→∞	n→∞	X
ejpam-4412	223	1	ar/2xn	ar/2xn	PROPN
ejpam-4412	223	2	exists	exist	VERB
ejpam-4412	223	3	,	,	PUNCT
ejpam-4412	223	4	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	223	5	,	,	PUNCT
ejpam-4412	223	6	n.	n.	NOUN
ejpam-4412	223	7	h.	h.	PROPN
ejpam-4412	223	8	altaweel	altaweel	PROPN
ejpam-4412	223	9	/	/	SYM
ejpam-4412	223	10	eur	eur	PROPN
ejpam-4412	223	11	.	.	PUNCT
ejpam-4412	224	1	j.	j.	PROPN
ejpam-4412	224	2	pure	pure	PROPN
ejpam-4412	224	3	appl	appl	PROPN
ejpam-4412	224	4	.	.	PROPN
ejpam-4412	224	5	math	math	PROPN
ejpam-4412	224	6	,	,	PUNCT
ejpam-4412	224	7	15	15	NUM
ejpam-4412	224	8	(	(	PUNCT
ejpam-4412	224	9	3	3	NUM
ejpam-4412	224	10	)	)	PUNCT
ejpam-4412	224	11	(	(	PUNCT
ejpam-4412	224	12	2022	2022	NUM
ejpam-4412	224	13	)	)	PUNCT
ejpam-4412	224	14	,	,	PUNCT
ejpam-4412	224	15	1067	1067	NUM
ejpam-4412	224	16	-	-	SYM
ejpam-4412	224	17	1089	1089	NUM
ejpam-4412	224	18	1075	1075	NUM
ejpam-4412	224	19	then	then	ADV
ejpam-4412	224	20	lim	lim	PROPN
ejpam-4412	224	21	n→∞	n→∞	X
ejpam-4412	225	1	ar/2xn	ar/2xn	X
ejpam-4412	225	2	=	=	SYM
ejpam-4412	225	3	0	0	NUM
ejpam-4412	225	4	for	for	ADP
ejpam-4412	225	5	any	any	DET
ejpam-4412	225	6	sequence	sequence	NOUN
ejpam-4412	225	7	of	of	ADP
ejpam-4412	225	8	vectors	vector	NOUN
ejpam-4412	225	9	{	{	PUNCT
ejpam-4412	225	10	xn	xn	NOUN
ejpam-4412	225	11	}	}	PUNCT
ejpam-4412	225	12	.	.	PUNCT
ejpam-4412	226	1	(	(	PUNCT
ejpam-4412	226	2	11	11	NUM
ejpam-4412	226	3	)	)	PUNCT
ejpam-4412	226	4	since	since	SCONJ
ejpam-4412	226	5	lim	lim	PROPN
ejpam-4412	226	6	n→∞	n→∞	X
ejpam-4412	226	7	br/2xn	br/2xn	PROPN
ejpam-4412	226	8	=	=	SYM
ejpam-4412	226	9	0	0	PUNCT
ejpam-4412	226	10	and	and	CCONJ
ejpam-4412	226	11	lim	lim	PROPN
ejpam-4412	226	12	n→∞	n→∞	X
ejpam-4412	227	1	ar/2xn	ar/2xn	PROPN
ejpam-4412	227	2	exists	exist	VERB
ejpam-4412	227	3	,	,	PUNCT
ejpam-4412	227	4	then	then	ADV
ejpam-4412	227	5	lim	lim	PROPN
ejpam-4412	227	6	n→∞	n→∞	PRON
ejpam-4412	227	7	b1/2xn	b1/2xn	PROPN
ejpam-4412	227	8	=	=	SYM
ejpam-4412	227	9	b(1−r)/2	b(1−r)/2	PROPN
ejpam-4412	227	10	(	(	PUNCT
ejpam-4412	227	11	lim	lim	PROPN
ejpam-4412	227	12	n→∞	n→∞	NUM
ejpam-4412	227	13	br/2xn	br/2xn	PROPN
ejpam-4412	227	14	)	)	PUNCT
ejpam-4412	227	15	=	=	SYM
ejpam-4412	227	16	0	0	PUNCT
ejpam-4412	228	1	and	and	CCONJ
ejpam-4412	228	2	lim	lim	PROPN
ejpam-4412	228	3	n→∞	n→∞	X
ejpam-4412	228	4	a1/2xn	a1/2xn	PROPN
ejpam-4412	228	5	=	=	SYM
ejpam-4412	228	6	a(1−r)/2	a(1−r)/2	PROPN
ejpam-4412	228	7	(	(	PUNCT
ejpam-4412	228	8	lim	lim	PROPN
ejpam-4412	228	9	n→∞	n→∞	X
ejpam-4412	228	10	ar/2xn	ar/2xn	NUM
ejpam-4412	228	11	)	)	PUNCT
ejpam-4412	228	12	exists	exist	VERB
ejpam-4412	228	13	,	,	PUNCT
ejpam-4412	228	14	so	so	SCONJ
ejpam-4412	228	15	that	that	SCONJ
ejpam-4412	228	16	lim	lim	PROPN
ejpam-4412	228	17	n→∞	n→∞	X
ejpam-4412	229	1	a1/2xn	a1/2xn	PROPN
ejpam-4412	229	2	=	=	SYM
ejpam-4412	229	3	0	0	NUM
ejpam-4412	229	4	by	by	ADP
ejpam-4412	229	5	(	(	PUNCT
ejpam-4412	229	6	8)	8)	NUM
ejpam-4412	229	7	,	,	PUNCT
ejpam-4412	229	8	hence	hence	ADV
ejpam-4412	229	9	lim	lim	NOUN
ejpam-4412	229	10	n→∞	n→∞	X
ejpam-4412	230	1	ar/2xn	ar/2xn	X
ejpam-4412	230	2	=	=	SYM
ejpam-4412	230	3	0	0	NUM
ejpam-4412	230	4	by	by	ADP
ejpam-4412	230	5	lemma	lemma	PROPN
ejpam-4412	230	6	4	4	NUM
ejpam-4412	230	7	.	.	PUNCT
ejpam-4412	230	8	(	(	PUNCT
ejpam-4412	230	9	11	11	NUM
ejpam-4412	230	10	)	)	PUNCT
ejpam-4412	230	11	ensures	ensure	VERB
ejpam-4412	230	12	ran(x	ran(x	X
ejpam-4412	230	13	)	)	PUNCT
ejpam-4412	230	14	=	=	SYM
ejpam-4412	230	15	ran(ar/2	ran(ar/2	X
ejpam-4412	230	16	)	)	PUNCT
ejpam-4412	230	17	by	by	ADP
ejpam-4412	230	18	lemma	lemma	PROPN
ejpam-4412	230	19	3	3	NUM
ejpam-4412	230	20	,	,	PUNCT
ejpam-4412	230	21	hence	hence	ADV
ejpam-4412	230	22	we	we	PRON
ejpam-4412	230	23	have	have	AUX
ejpam-4412	230	24	ker((ar/2cpar/2	ker((ar/2cpar/2	NOUN
ejpam-4412	230	25	)	)	PUNCT
ejpam-4412	230	26	rq	rq	X
ejpam-4412	230	27	p+r	p+r	NUM
ejpam-4412	230	28	)	)	PUNCT
ejpam-4412	230	29	=	=	SYM
ejpam-4412	230	30	ker(ar/2cpar/2	ker(ar/2cpar/2	NOUN
ejpam-4412	230	31	)	)	PUNCT
ejpam-4412	230	32	⊇	⊇	PROPN
ejpam-4412	230	33	ker(ar/2	ker(ar/2	NOUN
ejpam-4412	230	34	)	)	PUNCT
ejpam-4412	230	35	=	=	SYM
ejpam-4412	230	36	ker(ar	ker(ar	NOUN
ejpam-4412	230	37	)	)	PUNCT
ejpam-4412	230	38	=	=	SYM
ejpam-4412	230	39	ker(aqr	ker(aqr	ADJ
ejpam-4412	230	40	)	)	PUNCT
ejpam-4412	230	41	=	=	SYM
ejpam-4412	230	42	ker(x∗	ker(x∗	PROPN
ejpam-4412	230	43	)	)	PUNCT
ejpam-4412	230	44	,	,	PUNCT
ejpam-4412	230	45	so	so	SCONJ
ejpam-4412	230	46	that	that	SCONJ
ejpam-4412	230	47	aqr	aqr	ADV
ejpam-4412	230	48	=	=	X
ejpam-4412	230	49	(	(	PUNCT
ejpam-4412	230	50	ar/2cpar/2	ar/2cpar/2	NOUN
ejpam-4412	230	51	)	)	PUNCT
ejpam-4412	230	52	rq	rq	VERB
ejpam-4412	230	53	p+r	p+r	NUM
ejpam-4412	230	54	=	=	SYM
ejpam-4412	230	55	0	0	NUM
ejpam-4412	230	56	holds	hold	VERB
ejpam-4412	230	57	on	on	ADP
ejpam-4412	230	58	ker(x∗	ker(x∗	PROPN
ejpam-4412	230	59	)	)	PUNCT
ejpam-4412	230	60	.	.	PUNCT
ejpam-4412	231	1	consequently	consequently	ADV
ejpam-4412	231	2	the	the	DET
ejpam-4412	231	3	proof	proof	NOUN
ejpam-4412	231	4	is	be	AUX
ejpam-4412	231	5	complete	complete	ADJ
ejpam-4412	231	6	since	since	SCONJ
ejpam-4412	231	7	h	h	NOUN
ejpam-4412	231	8	=	=	PUNCT
ejpam-4412	231	9	ran(x)⊕	ran(x)⊕	NOUN
ejpam-4412	231	10	ker(x∗	ker(x∗	NOUN
ejpam-4412	231	11	)	)	PUNCT
ejpam-4412	231	12	.	.	PUNCT
ejpam-4412	232	1	lemma	lemma	PROPN
ejpam-4412	232	2	5	5	NUM
ejpam-4412	232	3	.	.	PUNCT
ejpam-4412	233	1	(	(	PUNCT
ejpam-4412	233	2	[	[	X
ejpam-4412	233	3	26	26	NUM
ejpam-4412	233	4	]	]	PUNCT
ejpam-4412	233	5	)	)	PUNCT
ejpam-4412	233	6	let	let	AUX
ejpam-4412	233	7	t	t	NOUN
ejpam-4412	233	8	=	=	SYM
ejpam-4412	233	9	u	u	SYM
ejpam-4412	233	10	|t	|t	VERB
ejpam-4412	233	11	|	|	PROPN
ejpam-4412	233	12	∈	∈	PROPN
ejpam-4412	233	13	b(h	b(h	PROPN
ejpam-4412	233	14	)	)	PUNCT
ejpam-4412	233	15	be	be	VERB
ejpam-4412	233	16	the	the	DET
ejpam-4412	233	17	polar	polar	ADJ
ejpam-4412	233	18	decomposition	decomposition	NOUN
ejpam-4412	233	19	of	of	ADP
ejpam-4412	233	20	t	t	PROPN
ejpam-4412	233	21	.	.	PUNCT
ejpam-4412	234	1	then	then	ADV
ejpam-4412	234	2	t	t	PROPN
ejpam-4412	234	3	is	be	AUX
ejpam-4412	234	4	class	class	NOUN
ejpam-4412	234	5	p	p	NOUN
ejpam-4412	234	6	-	-	PUNCT
ejpam-4412	234	7	wa(s	wa(s	NUM
ejpam-4412	234	8	,	,	PUNCT
ejpam-4412	234	9	t	t	NOUN
ejpam-4412	234	10	)	)	PUNCT
ejpam-4412	234	11	if	if	SCONJ
ejpam-4412	235	1	and	and	CCONJ
ejpam-4412	235	2	only	only	ADV
ejpam-4412	235	3	if	if	SCONJ
ejpam-4412	235	4	|t	|t	PROPN
ejpam-4412	235	5	(	(	PUNCT
ejpam-4412	235	6	s	s	X
ejpam-4412	235	7	,	,	PUNCT
ejpam-4412	235	8	t)|	t)|	ADJ
ejpam-4412	235	9	2tp	2tp	NOUN
ejpam-4412	235	10	s+t	s+t	PROPN
ejpam-4412	235	11	≥	≥	NOUN
ejpam-4412	235	12	|t	|t	VERB
ejpam-4412	235	13	|2tp	|2tp	PROPN
ejpam-4412	235	14	and	and	CCONJ
ejpam-4412	235	15	|t	|t	VERB
ejpam-4412	235	16	|2sp	|2sp	PROPN
ejpam-4412	235	17	≥	≥	NUM
ejpam-4412	235	18	|(t	|(t	PROPN
ejpam-4412	235	19	(	(	PUNCT
ejpam-4412	235	20	s	s	PROPN
ejpam-4412	235	21	,	,	PUNCT
ejpam-4412	235	22	t))∗|	t))∗|	NOUN
ejpam-4412	235	23	2sp	2sp	NOUN
ejpam-4412	235	24	s+t	s+t	PROPN
ejpam-4412	235	25	.	.	PUNCT
ejpam-4412	236	1	lemma	lemma	PROPN
ejpam-4412	236	2	6	6	NUM
ejpam-4412	236	3	.	.	PUNCT
ejpam-4412	237	1	let	let	VERB
ejpam-4412	237	2	0	0	NUM
ejpam-4412	237	3	<	<	X
ejpam-4412	237	4	s	s	PROPN
ejpam-4412	237	5	,	,	PUNCT
ejpam-4412	237	6	t	t	PROPN
ejpam-4412	237	7	,	,	PUNCT
ejpam-4412	237	8	s	s	PART
ejpam-4412	237	9	+	+	NUM
ejpam-4412	237	10	t	t	X
ejpam-4412	237	11	≤	≤	NUM
ejpam-4412	237	12	1	1	NUM
ejpam-4412	237	13	and	and	CCONJ
ejpam-4412	237	14	0	0	NUM
ejpam-4412	237	15	<	<	X
ejpam-4412	237	16	p	p	X
ejpam-4412	237	17	≤	≤	NUM
ejpam-4412	237	18	1	1	NUM
ejpam-4412	237	19	.	.	PUNCT
ejpam-4412	238	1	let	let	AUX
ejpam-4412	238	2	t	t	PROPN
ejpam-4412	238	3	∈	∈	PROPN
ejpam-4412	238	4	b(h	b(h	PROPN
ejpam-4412	238	5	)	)	PUNCT
ejpam-4412	238	6	be	be	AUX
ejpam-4412	238	7	class	class	NOUN
ejpam-4412	238	8	p	p	NOUN
ejpam-4412	238	9	-	-	PUNCT
ejpam-4412	238	10	wa(s	wa(s	NUM
ejpam-4412	238	11	,	,	PUNCT
ejpam-4412	238	12	t	t	PROPN
ejpam-4412	238	13	)	)	PUNCT
ejpam-4412	238	14	and	and	CCONJ
ejpam-4412	238	15	let	let	VERB
ejpam-4412	238	16	m	m	PRON
ejpam-4412	238	17	an	an	DET
ejpam-4412	238	18	invariant	invariant	ADJ
ejpam-4412	238	19	subspace	subspace	NOUN
ejpam-4412	238	20	of	of	ADP
ejpam-4412	238	21	t	t	PROPN
ejpam-4412	238	22	.	.	PUNCT
ejpam-4412	239	1	then	then	ADV
ejpam-4412	239	2	the	the	DET
ejpam-4412	239	3	restriction	restriction	NOUN
ejpam-4412	239	4	t	t	PROPN
ejpam-4412	239	5	|m	|m	NOUN
ejpam-4412	239	6	is	be	AUX
ejpam-4412	239	7	also	also	ADV
ejpam-4412	239	8	class	class	NOUN
ejpam-4412	239	9	p	p	NOUN
ejpam-4412	239	10	-	-	PUNCT
ejpam-4412	239	11	wa(s	wa(s	NUM
ejpam-4412	239	12	,	,	PUNCT
ejpam-4412	239	13	t	t	PROPN
ejpam-4412	239	14	)	)	PUNCT
ejpam-4412	239	15	.	.	PUNCT
ejpam-4412	240	1	proof	proof	NOUN
ejpam-4412	240	2	.	.	PUNCT
ejpam-4412	241	1	let	let	VERB
ejpam-4412	241	2	t	t	NOUN
ejpam-4412	241	3	=	=	SYM
ejpam-4412	241	4	(	(	PUNCT
ejpam-4412	241	5	t1	t1	PROPN
ejpam-4412	241	6	s	s	PART
ejpam-4412	241	7	0	0	NUM
ejpam-4412	241	8	t2	t2	NOUN
ejpam-4412	241	9	)	)	PUNCT
ejpam-4412	241	10	on	on	ADP
ejpam-4412	241	11	h	h	NOUN
ejpam-4412	241	12	=	=	NOUN
ejpam-4412	241	13	m	m	PROPN
ejpam-4412	241	14	⊕	⊕	NOUN
ejpam-4412	241	15	m⊥	m⊥	NOUN
ejpam-4412	241	16	and	and	CCONJ
ejpam-4412	241	17	p	p	X
ejpam-4412	241	18	the	the	DET
ejpam-4412	241	19	orthogonal	orthogonal	ADJ
ejpam-4412	241	20	projection	projection	NOUN
ejpam-4412	241	21	onto	onto	ADP
ejpam-4412	241	22	m.	m.	NOUN
ejpam-4412	241	23	let	let	VERB
ejpam-4412	241	24	t0	t0	NOUN
ejpam-4412	241	25	:	:	PUNCT
ejpam-4412	242	1	=	=	SYM
ejpam-4412	242	2	tp	tp	X
ejpam-4412	242	3	=	=	PUNCT
ejpam-4412	242	4	ptp	ptp	PROPN
ejpam-4412	242	5	=	=	SYM
ejpam-4412	242	6	(	(	PUNCT
ejpam-4412	242	7	t1	t1	NOUN
ejpam-4412	242	8	0	0	NUM
ejpam-4412	242	9	0	0	NUM
ejpam-4412	242	10	0	0	NUM
ejpam-4412	242	11	)	)	PUNCT
ejpam-4412	242	12	.	.	PUNCT
ejpam-4412	243	1	then	then	ADV
ejpam-4412	243	2	|t0|2	|t0|2	SYM
ejpam-4412	243	3	t	t	NOUN
ejpam-4412	243	4	=	=	PUNCT
ejpam-4412	243	5	(	(	PUNCT
ejpam-4412	243	6	p	p	NOUN
ejpam-4412	243	7	|t	|t	PROPN
ejpam-4412	243	8	|2p	|2p	NUM
ejpam-4412	243	9	)	)	PUNCT
ejpam-4412	243	10	t	t	NOUN
ejpam-4412	243	11	≥	≥	NOUN
ejpam-4412	243	12	p	p	X
ejpam-4412	243	13	|t	|t	PROPN
ejpam-4412	243	14	|2tp	|2tp	PROPN
ejpam-4412	243	15	for	for	ADP
ejpam-4412	243	16	each	each	DET
ejpam-4412	243	17	0	0	NUM
ejpam-4412	243	18	<	<	X
ejpam-4412	243	19	t	t	X
ejpam-4412	243	20	≤	≤	NUM
ejpam-4412	243	21	1	1	NUM
ejpam-4412	243	22	by	by	ADP
ejpam-4412	243	23	hansen	hansen	PROPN
ejpam-4412	243	24	’s	’s	PROPN
ejpam-4412	243	25	inequality	inequality	NOUN
ejpam-4412	243	26	,	,	PUNCT
ejpam-4412	243	27	and	and	CCONJ
ejpam-4412	243	28	|t	|t	PROPN
ejpam-4412	243	29	∗|2	∗|2	PUNCT
ejpam-4412	244	1	=	=	PUNCT
ejpam-4412	244	2	tt	tt	PROPN
ejpam-4412	244	3	∗	∗	X
ejpam-4412	244	4	≥	≥	NUM
ejpam-4412	244	5	tpt	tpt	NOUN
ejpam-4412	244	6	∗	∗	NOUN
ejpam-4412	244	7	=	=	PUNCT
ejpam-4412	244	8	|t	|t	PROPN
ejpam-4412	244	9	∗	∗	NOUN
ejpam-4412	244	10	0	0	NUM
ejpam-4412	244	11	|2	|2	NUM
ejpam-4412	244	12	.	.	PUNCT
ejpam-4412	245	1	hence	hence	ADV
ejpam-4412	245	2	t	t	PROPN
ejpam-4412	245	3	is	be	AUX
ejpam-4412	245	4	class	class	NOUN
ejpam-4412	245	5	p	p	NOUN
ejpam-4412	245	6	-	-	PUNCT
ejpam-4412	245	7	a(s	a(s	PROPN
ejpam-4412	245	8	,	,	PUNCT
ejpam-4412	245	9	t	t	PROPN
ejpam-4412	245	10	)	)	PUNCT
ejpam-4412	245	11	⇐	⇐	ADJ
ejpam-4412	245	12	⇒	⇒	NOUN
ejpam-4412	245	13	|t	|t	VERB
ejpam-4412	245	14	∗|2tp	∗|2tp	PUNCT
ejpam-4412	245	15	≤	≤	NOUN
ejpam-4412	245	16	(	(	PUNCT
ejpam-4412	245	17	|t	|t	ADJ
ejpam-4412	246	1	∗|t|t	∗|t|t	PROPN
ejpam-4412	246	2	|2s|t	|2s|t	NOUN
ejpam-4412	246	3	∗|t	∗|t	NOUN
ejpam-4412	246	4	)	)	PUNCT
ejpam-4412	246	5	tp	tp	ADP
ejpam-4412	246	6	s+t	s+t	PROPN
ejpam-4412	246	7	=	=	AUX
ejpam-4412	246	8	⇒	⇒	NOUN
ejpam-4412	246	9	|t	|t	VERB
ejpam-4412	246	10	∗	∗	NOUN
ejpam-4412	246	11	0	0	NUM
ejpam-4412	246	12	|2tp	|2tp	PROPN
ejpam-4412	246	13	≤	≤	NOUN
ejpam-4412	246	14	(	(	PUNCT
ejpam-4412	246	15	|t	|t	NOUN
ejpam-4412	246	16	∗	∗	NOUN
ejpam-4412	246	17	0	0	PUNCT
ejpam-4412	247	1	|t|t	|t|t	ADJ
ejpam-4412	247	2	|2s|t	|2s|t	NOUN
ejpam-4412	247	3	∗	∗	NOUN
ejpam-4412	247	4	0	0	NUM
ejpam-4412	247	5	|t	|t	PROPN
ejpam-4412	247	6	)	)	PUNCT
ejpam-4412	247	7	tp	tp	ADP
ejpam-4412	247	8	s+t	s+t	PROPN
ejpam-4412	247	9	(	(	PUNCT
ejpam-4412	247	10	by	by	ADP
ejpam-4412	247	11	lemma	lemma	PROPN
ejpam-4412	247	12	2	2	NUM
ejpam-4412	247	13	)	)	PUNCT
ejpam-4412	247	14	=	=	NOUN
ejpam-4412	247	15	⇒	⇒	NOUN
ejpam-4412	247	16	|t	|t	VERB
ejpam-4412	247	17	∗	∗	NOUN
ejpam-4412	247	18	0	0	NUM
ejpam-4412	247	19	|2tp	|2tp	PROPN
ejpam-4412	247	20	≤	≤	NOUN
ejpam-4412	247	21	(	(	PUNCT
ejpam-4412	247	22	|t	|t	PROPN
ejpam-4412	247	23	∗	∗	NOUN
ejpam-4412	247	24	0	0	NUM
ejpam-4412	248	1	|t|t0|2s|t	|t|t0|2s|t	PROPN
ejpam-4412	248	2	∗	∗	NOUN
ejpam-4412	248	3	0	0	NUM
ejpam-4412	248	4	|t	|t	PROPN
ejpam-4412	248	5	)	)	PUNCT
ejpam-4412	248	6	tp	tp	ADP
ejpam-4412	248	7	s+t	s+t	PROPN
ejpam-4412	248	8	(	(	PUNCT
ejpam-4412	248	9	since	since	SCONJ
ejpam-4412	248	10	|t	|t	PROPN
ejpam-4412	248	11	∗	∗	NOUN
ejpam-4412	248	12	0	0	NUM
ejpam-4412	248	13	|t	|t	PROPN
ejpam-4412	248	14	=	=	PUNCT
ejpam-4412	248	15	|t	|t	PROPN
ejpam-4412	248	16	∗	∗	NOUN
ejpam-4412	248	17	0	0	NUM
ejpam-4412	248	18	|tp	|tp	PRON
ejpam-4412	248	19	=	=	PUNCT
ejpam-4412	248	20	p	p	NOUN
ejpam-4412	248	21	|t	|t	PROPN
ejpam-4412	248	22	∗	∗	NOUN
ejpam-4412	248	23	0	0	NUM
ejpam-4412	248	24	|t	|t	PROPN
ejpam-4412	248	25	for	for	ADP
ejpam-4412	248	26	every	every	DET
ejpam-4412	248	27	0	0	NUM
ejpam-4412	248	28	<	<	X
ejpam-4412	248	29	t	t	X
ejpam-4412	248	30	≤	≤	NUM
ejpam-4412	248	31	1	1	NUM
ejpam-4412	248	32	)	)	PUNCT
ejpam-4412	248	33	.	.	PUNCT
ejpam-4412	249	1	now	now	ADV
ejpam-4412	249	2	|t0|	|t0|	X
ejpam-4412	249	3	=	=	PROPN
ejpam-4412	249	4	p	p	PROPN
ejpam-4412	249	5	|t̃	|t̃	PROPN
ejpam-4412	249	6	|p	|p	NOUN
ejpam-4412	249	7	≥	≥	PUNCT
ejpam-4412	249	8	p	p	X
ejpam-4412	249	9	|t	|t	PROPN
ejpam-4412	249	10	|p	|p	PROPN
ejpam-4412	249	11	≥	≥	X
ejpam-4412	249	12	p	p	X
ejpam-4412	249	13	|(t̃	|(t̃	X
ejpam-4412	249	14	)	)	PUNCT
ejpam-4412	249	15	∗|p	∗|p	PROPN
ejpam-4412	249	16	=	=	SYM
ejpam-4412	249	17	|t	|t	PROPN
ejpam-4412	249	18	∗	∗	NOUN
ejpam-4412	249	19	0	0	NUM
ejpam-4412	250	1	|	|	ADV
ejpam-4412	250	2	.	.	PUNCT
ejpam-4412	251	1	then	then	ADV
ejpam-4412	251	2	by	by	ADP
ejpam-4412	251	3	theorem	theorem	NOUN
ejpam-4412	251	4	3	3	NUM
ejpam-4412	251	5	it	it	PRON
ejpam-4412	251	6	follows	follow	VERB
ejpam-4412	251	7	that	that	SCONJ
ejpam-4412	251	8	|t0|2sp	|t0|2sp	NOUN
ejpam-4412	251	9	≥	≥	NOUN
ejpam-4412	251	10	(	(	PUNCT
ejpam-4412	251	11	|t0|s|t	|t0|s|t	NOUN
ejpam-4412	251	12	∗	∗	NOUN
ejpam-4412	251	13	0	0	NUM
ejpam-4412	251	14	|2t||t0|s|	|2t||t0|s|	NOUN
ejpam-4412	251	15	)	)	PUNCT
ejpam-4412	251	16	ps	ps	NOUN
ejpam-4412	251	17	s+t	s+t	PROPN
ejpam-4412	251	18	.	.	PUNCT
ejpam-4412	252	1	therefore	therefore	ADV
ejpam-4412	252	2	,	,	PUNCT
ejpam-4412	252	3	t	t	PROPN
ejpam-4412	252	4	|m	|m	NOUN
ejpam-4412	252	5	is	be	AUX
ejpam-4412	252	6	class	class	NOUN
ejpam-4412	252	7	p	p	NOUN
ejpam-4412	252	8	-	-	PUNCT
ejpam-4412	252	9	a(s	a(s	PROPN
ejpam-4412	252	10	,	,	PUNCT
ejpam-4412	252	11	t	t	PROPN
ejpam-4412	252	12	)	)	PUNCT
ejpam-4412	252	13	operator	operator	NOUN
ejpam-4412	252	14	.	.	PUNCT
ejpam-4412	253	1	the	the	DET
ejpam-4412	253	2	following	follow	VERB
ejpam-4412	253	3	example	example	NOUN
ejpam-4412	253	4	shows	show	VERB
ejpam-4412	253	5	that	that	SCONJ
ejpam-4412	253	6	there	there	PRON
ejpam-4412	253	7	exists	exist	VERB
ejpam-4412	253	8	a	a	DET
ejpam-4412	253	9	class	class	NOUN
ejpam-4412	253	10	p	p	NOUN
ejpam-4412	253	11	-	-	PUNCT
ejpam-4412	253	12	wa(s	wa(s	NUM
ejpam-4412	253	13	,	,	PUNCT
ejpam-4412	253	14	t	t	NOUN
ejpam-4412	253	15	)	)	PUNCT
ejpam-4412	253	16	operator	operator	NOUN
ejpam-4412	253	17	t	t	NOUN
ejpam-4412	253	18	such	such	ADJ
ejpam-4412	253	19	that	that	SCONJ
ejpam-4412	253	20	t	t	PROPN
ejpam-4412	253	21	|m	|m	NOUN
ejpam-4412	253	22	is	be	AUX
ejpam-4412	253	23	quasinormal	quasinormal	ADJ
ejpam-4412	253	24	but	but	CCONJ
ejpam-4412	253	25	m	m	PRON
ejpam-4412	253	26	does	do	AUX
ejpam-4412	253	27	not	not	PART
ejpam-4412	253	28	reduce	reduce	VERB
ejpam-4412	253	29	t	t	NOUN
ejpam-4412	253	30	.	.	PUNCT
ejpam-4412	254	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	254	2	,	,	PUNCT
ejpam-4412	254	3	n.	n.	NOUN
ejpam-4412	254	4	h.	h.	PROPN
ejpam-4412	254	5	altaweel	altaweel	PROPN
ejpam-4412	254	6	/	/	SYM
ejpam-4412	254	7	eur	eur	PROPN
ejpam-4412	254	8	.	.	PUNCT
ejpam-4412	255	1	j.	j.	PROPN
ejpam-4412	255	2	pure	pure	PROPN
ejpam-4412	255	3	appl	appl	PROPN
ejpam-4412	255	4	.	.	PROPN
ejpam-4412	255	5	math	math	PROPN
ejpam-4412	255	6	,	,	PUNCT
ejpam-4412	255	7	15	15	NUM
ejpam-4412	255	8	(	(	PUNCT
ejpam-4412	255	9	3	3	NUM
ejpam-4412	255	10	)	)	PUNCT
ejpam-4412	255	11	(	(	PUNCT
ejpam-4412	255	12	2022	2022	NUM
ejpam-4412	255	13	)	)	PUNCT
ejpam-4412	255	14	,	,	PUNCT
ejpam-4412	255	15	1067	1067	NUM
ejpam-4412	255	16	-	-	SYM
ejpam-4412	255	17	1089	1089	NUM
ejpam-4412	255	18	1076	1076	NUM
ejpam-4412	255	19	example	example	NOUN
ejpam-4412	255	20	1	1	NUM
ejpam-4412	255	21	.	.	PUNCT
ejpam-4412	256	1	let	let	VERB
ejpam-4412	256	2	t	t	PROPN
ejpam-4412	256	3	be	be	AUX
ejpam-4412	256	4	a	a	DET
ejpam-4412	256	5	bilateral	bilateral	ADJ
ejpam-4412	256	6	shift	shift	NOUN
ejpam-4412	256	7	on	on	ADP
ejpam-4412	256	8	ℓ2(z	ℓ2(z	NOUN
ejpam-4412	256	9	)	)	PUNCT
ejpam-4412	256	10	defined	define	VERB
ejpam-4412	256	11	by	by	ADP
ejpam-4412	256	12	ten	ten	NUM
ejpam-4412	256	13	=	=	SYM
ejpam-4412	256	14	en+1	en+1	PROPN
ejpam-4412	256	15	and	and	CCONJ
ejpam-4412	256	16	m	m	VERB
ejpam-4412	256	17	=	=	NOUN
ejpam-4412	256	18	∨	∨	NUM
ejpam-4412	256	19	n≥0	n≥0	PROPN
ejpam-4412	256	20	cen	cen	PROPN
ejpam-4412	256	21	.	.	PUNCT
ejpam-4412	257	1	then	then	ADV
ejpam-4412	257	2	t	t	PROPN
ejpam-4412	257	3	is	be	AUX
ejpam-4412	257	4	unitary	unitary	ADJ
ejpam-4412	257	5	and	and	CCONJ
ejpam-4412	257	6	t	t	NOUN
ejpam-4412	257	7	|m	|m	NOUN
ejpam-4412	257	8	is	be	AUX
ejpam-4412	257	9	isometry	isometry	ADJ
ejpam-4412	257	10	.	.	PUNCT
ejpam-4412	258	1	however	however	ADV
ejpam-4412	258	2	,	,	PUNCT
ejpam-4412	258	3	m	m	VERB
ejpam-4412	258	4	does	do	AUX
ejpam-4412	258	5	not	not	PART
ejpam-4412	258	6	reduce	reduce	VERB
ejpam-4412	258	7	t.	t.	PROPN
ejpam-4412	258	8	lemma	lemma	PROPN
ejpam-4412	258	9	7	7	X
ejpam-4412	258	10	.	.	PUNCT
ejpam-4412	259	1	let	let	VERB
ejpam-4412	259	2	0	0	NUM
ejpam-4412	259	3	<	<	X
ejpam-4412	259	4	s	s	PROPN
ejpam-4412	259	5	,	,	PUNCT
ejpam-4412	259	6	t	t	PROPN
ejpam-4412	259	7	,	,	PUNCT
ejpam-4412	259	8	s+t	s+t	PROPN
ejpam-4412	259	9	=	=	SYM
ejpam-4412	259	10	1	1	NUM
ejpam-4412	259	11	and	and	CCONJ
ejpam-4412	259	12	0	0	NUM
ejpam-4412	259	13	<	<	X
ejpam-4412	259	14	p	p	X
ejpam-4412	259	15	≤	≤	NUM
ejpam-4412	259	16	1	1	NUM
ejpam-4412	259	17	.	.	PUNCT
ejpam-4412	260	1	let	let	AUX
ejpam-4412	260	2	t	t	PROPN
ejpam-4412	260	3	∈	∈	PROPN
ejpam-4412	260	4	b(h	b(h	PROPN
ejpam-4412	260	5	)	)	PUNCT
ejpam-4412	260	6	be	be	AUX
ejpam-4412	260	7	class	class	NOUN
ejpam-4412	260	8	p	p	NOUN
ejpam-4412	260	9	-	-	PUNCT
ejpam-4412	260	10	wa(s	wa(s	NUM
ejpam-4412	260	11	,	,	PUNCT
ejpam-4412	260	12	t	t	NOUN
ejpam-4412	260	13	)	)	PUNCT
ejpam-4412	260	14	operator	operator	NOUN
ejpam-4412	260	15	,	,	PUNCT
ejpam-4412	260	16	let	let	VERB
ejpam-4412	260	17	m	m	PRON
ejpam-4412	260	18	be	be	AUX
ejpam-4412	260	19	an	an	DET
ejpam-4412	260	20	invariant	invariant	ADJ
ejpam-4412	260	21	subspace	subspace	NOUN
ejpam-4412	260	22	for	for	ADP
ejpam-4412	260	23	t	t	PROPN
ejpam-4412	260	24	and	and	CCONJ
ejpam-4412	260	25	a	a	DET
ejpam-4412	260	26	reducing	reduce	VERB
ejpam-4412	260	27	subspace	subspace	NOUN
ejpam-4412	260	28	for	for	ADP
ejpam-4412	260	29	t	t	PROPN
ejpam-4412	260	30	(	(	PUNCT
ejpam-4412	260	31	s	s	PROPN
ejpam-4412	260	32	,	,	PUNCT
ejpam-4412	260	33	t	t	PROPN
ejpam-4412	260	34	)	)	PUNCT
ejpam-4412	260	35	such	such	ADJ
ejpam-4412	260	36	that	that	SCONJ
ejpam-4412	260	37	t	t	PROPN
ejpam-4412	260	38	(	(	PUNCT
ejpam-4412	260	39	s	s	PROPN
ejpam-4412	260	40	,	,	PUNCT
ejpam-4412	260	41	t)|m	t)|m	VERB
ejpam-4412	260	42	the	the	DET
ejpam-4412	260	43	restriction	restriction	NOUN
ejpam-4412	260	44	of	of	ADP
ejpam-4412	260	45	t	t	PROPN
ejpam-4412	260	46	(	(	PUNCT
ejpam-4412	260	47	s	s	PROPN
ejpam-4412	260	48	,	,	PUNCT
ejpam-4412	260	49	t	t	PROPN
ejpam-4412	260	50	)	)	PUNCT
ejpam-4412	260	51	to	to	ADP
ejpam-4412	260	52	m	m	PROPN
ejpam-4412	260	53	is	be	AUX
ejpam-4412	260	54	an	an	DET
ejpam-4412	260	55	injective	injective	ADJ
ejpam-4412	260	56	normal	normal	ADJ
ejpam-4412	260	57	operator	operator	NOUN
ejpam-4412	260	58	,	,	PUNCT
ejpam-4412	260	59	then	then	ADV
ejpam-4412	260	60	t	t	VERB
ejpam-4412	260	61	|m	|m	NOUN
ejpam-4412	260	62	=	=	SYM
ejpam-4412	260	63	t	t	PROPN
ejpam-4412	260	64	(	(	PUNCT
ejpam-4412	260	65	s	s	X
ejpam-4412	260	66	,	,	PUNCT
ejpam-4412	260	67	t)|m	t)|m	NOUN
ejpam-4412	260	68	and	and	CCONJ
ejpam-4412	260	69	m	m	VERB
ejpam-4412	260	70	reduces	reduce	VERB
ejpam-4412	260	71	t.	t.	NOUN
ejpam-4412	260	72	proof	proof	NOUN
ejpam-4412	260	73	.	.	PUNCT
ejpam-4412	261	1	let	let	VERB
ejpam-4412	261	2	t	t	PROPN
ejpam-4412	261	3	(	(	PUNCT
ejpam-4412	261	4	s	s	PROPN
ejpam-4412	261	5	,	,	PUNCT
ejpam-4412	261	6	t	t	PROPN
ejpam-4412	261	7	)	)	PUNCT
ejpam-4412	261	8	=	=	PUNCT
ejpam-4412	262	1	(	(	PUNCT
ejpam-4412	262	2	t0	t0	NOUN
ejpam-4412	262	3	0	0	NUM
ejpam-4412	262	4	0	0	NUM
ejpam-4412	262	5	a	a	PRON
ejpam-4412	262	6	)	)	PUNCT
ejpam-4412	262	7	,	,	PUNCT
ejpam-4412	262	8	t	t	NOUN
ejpam-4412	262	9	=	=	PUNCT
ejpam-4412	262	10	(	(	PUNCT
ejpam-4412	262	11	s	s	PROPN
ejpam-4412	262	12	b	b	NOUN
ejpam-4412	262	13	0	0	NUM
ejpam-4412	262	14	d	d	NOUN
ejpam-4412	262	15	)	)	PUNCT
ejpam-4412	262	16	on	on	ADP
ejpam-4412	262	17	h	h	NOUN
ejpam-4412	262	18	=	=	PUNCT
ejpam-4412	262	19	m⊕m⊥.	m⊕m⊥.	PUNCT
ejpam-4412	262	20	since	since	SCONJ
ejpam-4412	262	21	t	t	PROPN
ejpam-4412	262	22	is	be	AUX
ejpam-4412	262	23	class	class	NOUN
ejpam-4412	262	24	p	p	NOUN
ejpam-4412	262	25	-	-	PUNCT
ejpam-4412	262	26	wa(s	wa(s	NUM
ejpam-4412	262	27	,	,	PUNCT
ejpam-4412	262	28	t	t	PROPN
ejpam-4412	262	29	)	)	PUNCT
ejpam-4412	262	30	we	we	PRON
ejpam-4412	262	31	have	have	AUX
ejpam-4412	262	32	|t	|t	VERB
ejpam-4412	262	33	(	(	PUNCT
ejpam-4412	262	34	s	s	PROPN
ejpam-4412	262	35	,	,	PUNCT
ejpam-4412	262	36	t)|2rp	t)|2rp	ADJ
ejpam-4412	262	37	≥	≥	NOUN
ejpam-4412	262	38	|t	|t	ADV
ejpam-4412	263	1	|2rp	|2rp	ADJ
ejpam-4412	263	2	≥	≥	NUM
ejpam-4412	263	3	|(t	|(t	PROPN
ejpam-4412	263	4	(	(	PUNCT
ejpam-4412	263	5	s	s	PROPN
ejpam-4412	263	6	,	,	PUNCT
ejpam-4412	263	7	t))∗|2rp	t))∗|2rp	VERB
ejpam-4412	263	8	for	for	ADP
ejpam-4412	263	9	r	r	NOUN
ejpam-4412	263	10	∈	∈	PROPN
ejpam-4412	263	11	min{s	min{s	PROPN
ejpam-4412	263	12	,	,	PUNCT
ejpam-4412	263	13	t	t	PROPN
ejpam-4412	263	14	}	}	PUNCT
ejpam-4412	263	15	.	.	PUNCT
ejpam-4412	264	1	let	let	VERB
ejpam-4412	264	2	p	p	PRON
ejpam-4412	264	3	be	be	AUX
ejpam-4412	264	4	the	the	DET
ejpam-4412	264	5	orthogonal	orthogonal	ADJ
ejpam-4412	264	6	projection	projection	NOUN
ejpam-4412	264	7	onto	onto	ADP
ejpam-4412	264	8	m.	m.	NOUN
ejpam-4412	264	9	then	then	ADV
ejpam-4412	264	10	|t0|	|t0|	X
ejpam-4412	265	1	=	=	PROPN
ejpam-4412	265	2	p	p	X
ejpam-4412	265	3	|t	|t	PROPN
ejpam-4412	265	4	(	(	PUNCT
ejpam-4412	265	5	s	s	PROPN
ejpam-4412	265	6	,	,	PUNCT
ejpam-4412	265	7	t)|p	t)|p	PRON
ejpam-4412	265	8	≥	≥	PROPN
ejpam-4412	265	9	p	p	X
ejpam-4412	265	10	|t	|t	PROPN
ejpam-4412	265	11	|p	|p	PROPN
ejpam-4412	265	12	≥	≥	X
ejpam-4412	265	13	p	p	X
ejpam-4412	265	14	|(t	|(t	PROPN
ejpam-4412	265	15	(	(	PUNCT
ejpam-4412	265	16	s	s	PROPN
ejpam-4412	265	17	,	,	PUNCT
ejpam-4412	265	18	t))∗|p	t))∗|p	PUNCT
ejpam-4412	265	19	=	=	PRON
ejpam-4412	265	20	|t	|t	PROPN
ejpam-4412	265	21	∗	∗	NOUN
ejpam-4412	265	22	0	0	NUM
ejpam-4412	266	1	|	|	ADV
ejpam-4412	266	2	.	.	PUNCT
ejpam-4412	267	1	by	by	ADP
ejpam-4412	267	2	löwner	löwner	NOUN
ejpam-4412	267	3	-	-	PUNCT
ejpam-4412	267	4	heinz	heinz	NOUN
ejpam-4412	267	5	theorem	theorem	NOUN
ejpam-4412	267	6	we	we	PRON
ejpam-4412	267	7	get	get	VERB
ejpam-4412	267	8	|t0|2rp	|t0|2rp	ADJ
ejpam-4412	268	1	=	=	SYM
ejpam-4412	268	2	p	p	X
ejpam-4412	268	3	|t	|t	PROPN
ejpam-4412	268	4	(	(	PUNCT
ejpam-4412	268	5	s	s	PROPN
ejpam-4412	268	6	,	,	PUNCT
ejpam-4412	268	7	t)|2rpp	t)|2rpp	PRON
ejpam-4412	268	8	≥	≥	NOUN
ejpam-4412	269	1	p	p	X
ejpam-4412	269	2	|t	|t	PROPN
ejpam-4412	269	3	|2rpp	|2rpp	X
ejpam-4412	269	4	≥	≥	NOUN
ejpam-4412	269	5	p	p	X
ejpam-4412	269	6	|(t	|(t	PROPN
ejpam-4412	269	7	(	(	PUNCT
ejpam-4412	269	8	s	s	PROPN
ejpam-4412	269	9	,	,	PUNCT
ejpam-4412	269	10	t))∗|2rpp	t))∗|2rpp	PUNCT
ejpam-4412	269	11	=	=	PRON
ejpam-4412	269	12	|t	|t	PROPN
ejpam-4412	269	13	∗	∗	NOUN
ejpam-4412	269	14	0	0	NUM
ejpam-4412	269	15	|2rp	|2rp	NOUN
ejpam-4412	269	16	.	.	PUNCT
ejpam-4412	270	1	since	since	SCONJ
ejpam-4412	270	2	|t	|t	PROPN
ejpam-4412	270	3	|st	|st	PROPN
ejpam-4412	270	4	=	=	SYM
ejpam-4412	270	5	t	t	PROPN
ejpam-4412	270	6	(	(	PUNCT
ejpam-4412	270	7	s	s	PROPN
ejpam-4412	270	8	,	,	PUNCT
ejpam-4412	270	9	t)|t	t)|t	NOUN
ejpam-4412	270	10	|s	|s	PROPN
ejpam-4412	270	11	and	and	CCONJ
ejpam-4412	270	12	p	p	NOUN
ejpam-4412	270	13	|t	|t	PROPN
ejpam-4412	270	14	|sp	|sp	PROPN
ejpam-4412	270	15	=	=	SYM
ejpam-4412	270	16	|t0|s	|t0|s	PROPN
ejpam-4412	270	17	,	,	PUNCT
ejpam-4412	270	18	we	we	PRON
ejpam-4412	270	19	deduce	deduce	VERB
ejpam-4412	270	20	that	that	SCONJ
ejpam-4412	270	21	|t0|ss	|t0|ss	PROPN
ejpam-4412	270	22	=	=	PUNCT
ejpam-4412	271	1	t0|t0|s	t0|t0|s	NOUN
ejpam-4412	271	2	.	.	PUNCT
ejpam-4412	272	1	we	we	PRON
ejpam-4412	272	2	have	have	VERB
ejpam-4412	272	3	t0	t0	PROPN
ejpam-4412	272	4	is	be	AUX
ejpam-4412	272	5	an	an	DET
ejpam-4412	272	6	injective	injective	ADJ
ejpam-4412	272	7	normal	normal	ADJ
ejpam-4412	272	8	operator	operator	NOUN
ejpam-4412	272	9	,	,	PUNCT
ejpam-4412	272	10	then	then	ADV
ejpam-4412	272	11	s	s	PART
ejpam-4412	272	12	=	=	SYM
ejpam-4412	272	13	t	t	X
ejpam-4412	272	14	|m	|m	NOUN
ejpam-4412	272	15	=	=	SYM
ejpam-4412	272	16	t0	t0	PROPN
ejpam-4412	272	17	=	=	SYM
ejpam-4412	272	18	t	t	PROPN
ejpam-4412	272	19	(	(	PUNCT
ejpam-4412	272	20	s	s	X
ejpam-4412	272	21	,	,	PUNCT
ejpam-4412	272	22	t)|m	t)|m	PROPN
ejpam-4412	272	23	,	,	PUNCT
ejpam-4412	272	24	consequently	consequently	ADV
ejpam-4412	272	25	t	t	PROPN
ejpam-4412	272	26	=	=	SYM
ejpam-4412	272	27	(	(	PUNCT
ejpam-4412	272	28	t0	t0	PROPN
ejpam-4412	272	29	b	b	PROPN
ejpam-4412	272	30	0	0	NUM
ejpam-4412	272	31	d	d	NOUN
ejpam-4412	272	32	)	)	PUNCT
ejpam-4412	272	33	on	on	ADP
ejpam-4412	272	34	h	h	NOUN
ejpam-4412	272	35	=	=	PUNCT
ejpam-4412	272	36	m⊕m⊥.	m⊕m⊥.	PUNCT
ejpam-4412	272	37	hence	hence	ADV
ejpam-4412	272	38	t	t	X
ejpam-4412	272	39	∗t	∗t	PROPN
ejpam-4412	272	40	=	=	SYM
ejpam-4412	272	41	(	(	PUNCT
ejpam-4412	272	42	t	t	PROPN
ejpam-4412	272	43	∗	∗	NOUN
ejpam-4412	272	44	0	0	NUM
ejpam-4412	272	45	t0	t0	PROPN
ejpam-4412	272	46	t	t	PROPN
ejpam-4412	272	47	∗	∗	NOUN
ejpam-4412	272	48	0b	0b	PROPN
ejpam-4412	272	49	b∗t0	b∗t0	PROPN
ejpam-4412	272	50	b∗b	b∗b	X
ejpam-4412	273	1	+	+	ADJ
ejpam-4412	273	2	d∗d	d∗d	NUM
ejpam-4412	273	3	)	)	PUNCT
ejpam-4412	273	4	on	on	ADP
ejpam-4412	273	5	h	h	NOUN
ejpam-4412	273	6	=	=	PRON
ejpam-4412	273	7	m⊕m⊥.	m⊕m⊥.	PUNCT
ejpam-4412	274	1	so	so	ADV
ejpam-4412	274	2	we	we	PRON
ejpam-4412	274	3	can	can	AUX
ejpam-4412	274	4	write	write	VERB
ejpam-4412	274	5	|t	|t	PROPN
ejpam-4412	274	6	|rp	|rp	NUM
ejpam-4412	274	7	=	=	SYM
ejpam-4412	274	8	(	(	PUNCT
ejpam-4412	274	9	|t0|rp	|t0|rp	PROPN
ejpam-4412	274	10	x	x	SYM
ejpam-4412	274	11	x∗	x∗	PROPN
ejpam-4412	274	12	y	y	PROPN
ejpam-4412	274	13	)	)	PUNCT
ejpam-4412	274	14	on	on	ADP
ejpam-4412	274	15	h	h	NOUN
ejpam-4412	274	16	=	=	PUNCT
ejpam-4412	274	17	m⊕m⊥.	m⊕m⊥.	PUNCT
ejpam-4412	274	18	since	since	SCONJ
ejpam-4412	274	19	p	p	PRON
ejpam-4412	274	20	|t	|t	ADJ
ejpam-4412	274	21	|pr|t	|pr|t	X
ejpam-4412	274	22	|prp	|prp	PROPN
ejpam-4412	274	23	=	=	SYM
ejpam-4412	274	24	|t0|2rp	|t0|2rp	NOUN
ejpam-4412	274	25	,	,	PUNCT
ejpam-4412	274	26	then	then	ADV
ejpam-4412	274	27	|t0|2rp	|t0|2rp	VERB
ejpam-4412	274	28	=	=	SYM
ejpam-4412	274	29	|t0|2rp	|t0|2rp	VERB
ejpam-4412	274	30	+	+	NOUN
ejpam-4412	274	31	xx∗	xx∗	NOUN
ejpam-4412	274	32	,	,	PUNCT
ejpam-4412	274	33	and	and	CCONJ
ejpam-4412	274	34	thus	thus	ADV
ejpam-4412	274	35	x	x	X
ejpam-4412	274	36	=	=	NOUN
ejpam-4412	274	37	0	0	X
ejpam-4412	274	38	.	.	PUNCT
ejpam-4412	275	1	it	it	PRON
ejpam-4412	275	2	follows	follow	VERB
ejpam-4412	275	3	that	that	SCONJ
ejpam-4412	275	4	|t	|t	PROPN
ejpam-4412	275	5	|rp	|rp	NUM
ejpam-4412	275	6	=	=	PUNCT
ejpam-4412	275	7	|t0|rp⊕y	|t0|rp⊕y	NOUN
ejpam-4412	275	8	2	2	NUM
ejpam-4412	275	9	implying	imply	VERB
ejpam-4412	275	10	|t	|t	VERB
ejpam-4412	275	11	|2rp	|2rp	PROPN
ejpam-4412	275	12	=	=	SYM
ejpam-4412	275	13	|t0|2rp⊕y	|t0|2rp⊕y	PROPN
ejpam-4412	275	14	4	4	NUM
ejpam-4412	275	15	.	.	PUNCT
ejpam-4412	276	1	consequently	consequently	ADV
ejpam-4412	276	2	we	we	PRON
ejpam-4412	276	3	get	get	VERB
ejpam-4412	276	4	b∗b	b∗b	ADJ
ejpam-4412	276	5	=	=	SYM
ejpam-4412	276	6	0	0	NUM
ejpam-4412	277	1	it	it	PRON
ejpam-4412	277	2	follows	follow	VERB
ejpam-4412	277	3	that	that	PRON
ejpam-4412	277	4	b	b	NOUN
ejpam-4412	277	5	=	=	SYM
ejpam-4412	277	6	0	0	PUNCT
ejpam-4412	277	7	and	and	CCONJ
ejpam-4412	277	8	hence	hence	ADV
ejpam-4412	277	9	m	m	AUX
ejpam-4412	277	10	reduces	reduce	VERB
ejpam-4412	277	11	t	t	NOUN
ejpam-4412	277	12	.	.	PUNCT
ejpam-4412	278	1	the	the	DET
ejpam-4412	278	2	next	next	ADJ
ejpam-4412	278	3	lemma	lemma	PROPN
ejpam-4412	278	4	is	be	AUX
ejpam-4412	278	5	a	a	DET
ejpam-4412	278	6	simple	simple	ADJ
ejpam-4412	278	7	consequence	consequence	NOUN
ejpam-4412	278	8	of	of	ADP
ejpam-4412	278	9	the	the	DET
ejpam-4412	278	10	preceding	precede	VERB
ejpam-4412	278	11	one	one	NUM
ejpam-4412	278	12	.	.	PUNCT
ejpam-4412	279	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	279	2	,	,	PUNCT
ejpam-4412	279	3	n.	n.	NOUN
ejpam-4412	279	4	h.	h.	PROPN
ejpam-4412	279	5	altaweel	altaweel	PROPN
ejpam-4412	279	6	/	/	SYM
ejpam-4412	279	7	eur	eur	PROPN
ejpam-4412	279	8	.	.	PUNCT
ejpam-4412	280	1	j.	j.	PROPN
ejpam-4412	280	2	pure	pure	PROPN
ejpam-4412	280	3	appl	appl	PROPN
ejpam-4412	280	4	.	.	PROPN
ejpam-4412	280	5	math	math	PROPN
ejpam-4412	280	6	,	,	PUNCT
ejpam-4412	280	7	15	15	NUM
ejpam-4412	280	8	(	(	PUNCT
ejpam-4412	280	9	3	3	NUM
ejpam-4412	280	10	)	)	PUNCT
ejpam-4412	280	11	(	(	PUNCT
ejpam-4412	280	12	2022	2022	NUM
ejpam-4412	280	13	)	)	PUNCT
ejpam-4412	280	14	,	,	PUNCT
ejpam-4412	280	15	1067	1067	NUM
ejpam-4412	280	16	-	-	SYM
ejpam-4412	280	17	1089	1089	NUM
ejpam-4412	280	18	1077	1077	NUM
ejpam-4412	280	19	lemma	lemma	PROPN
ejpam-4412	280	20	8	8	NUM
ejpam-4412	280	21	.	.	PUNCT
ejpam-4412	281	1	let	let	VERB
ejpam-4412	281	2	0	0	NUM
ejpam-4412	281	3	<	<	X
ejpam-4412	281	4	s	s	PROPN
ejpam-4412	281	5	,	,	PUNCT
ejpam-4412	281	6	t	t	PROPN
ejpam-4412	281	7	,	,	PUNCT
ejpam-4412	281	8	s	s	PART
ejpam-4412	281	9	+	+	NUM
ejpam-4412	281	10	t	t	X
ejpam-4412	281	11	=	=	SYM
ejpam-4412	281	12	1	1	NUM
ejpam-4412	281	13	and	and	CCONJ
ejpam-4412	281	14	0	0	NUM
ejpam-4412	281	15	<	<	X
ejpam-4412	281	16	p	p	X
ejpam-4412	281	17	≤	≤	NUM
ejpam-4412	281	18	1	1	NUM
ejpam-4412	281	19	.	.	PUNCT
ejpam-4412	282	1	let	let	AUX
ejpam-4412	282	2	t	t	PROPN
ejpam-4412	282	3	∈	∈	PROPN
ejpam-4412	282	4	b(h	b(h	PROPN
ejpam-4412	282	5	)	)	PUNCT
ejpam-4412	282	6	be	be	VERB
ejpam-4412	282	7	a	a	DET
ejpam-4412	282	8	class	class	NOUN
ejpam-4412	282	9	p	p	NOUN
ejpam-4412	282	10	-	-	PUNCT
ejpam-4412	282	11	wa(s	wa(s	NUM
ejpam-4412	282	12	,	,	PUNCT
ejpam-4412	282	13	t	t	NOUN
ejpam-4412	282	14	)	)	PUNCT
ejpam-4412	282	15	operator	operator	NOUN
ejpam-4412	282	16	with	with	ADP
ejpam-4412	282	17	ker(t	ker(t	NOUN
ejpam-4412	282	18	)	)	PUNCT
ejpam-4412	283	1	⊂	⊂	PROPN
ejpam-4412	283	2	ker(t	ker(t	NOUN
ejpam-4412	283	3	∗	∗	NOUN
ejpam-4412	283	4	)	)	PUNCT
ejpam-4412	283	5	.	.	PUNCT
ejpam-4412	284	1	then	then	ADV
ejpam-4412	284	2	t	t	PROPN
ejpam-4412	284	3	=	=	SYM
ejpam-4412	284	4	t1	t1	PROPN
ejpam-4412	284	5	⊕	⊕	PROPN
ejpam-4412	284	6	t2	t2	PROPN
ejpam-4412	284	7	on	on	ADP
ejpam-4412	284	8	h	h	NOUN
ejpam-4412	284	9	=	=	PUNCT
ejpam-4412	284	10	h1	h1	PROPN
ejpam-4412	284	11	⊕h2	⊕h2	NOUN
ejpam-4412	284	12	where	where	SCONJ
ejpam-4412	284	13	t1	t1	NOUN
ejpam-4412	284	14	is	be	AUX
ejpam-4412	284	15	normal	normal	ADJ
ejpam-4412	284	16	,	,	PUNCT
ejpam-4412	284	17	ker(t2	ker(t2	NOUN
ejpam-4412	284	18	)	)	PUNCT
ejpam-4412	284	19	=	=	PUNCT
ejpam-4412	284	20	{	{	PUNCT
ejpam-4412	284	21	0	0	NUM
ejpam-4412	284	22	}	}	PUNCT
ejpam-4412	284	23	and	and	CCONJ
ejpam-4412	284	24	t2	t2	NOUN
ejpam-4412	284	25	is	be	AUX
ejpam-4412	284	26	pure	pure	ADJ
ejpam-4412	284	27	class	class	NOUN
ejpam-4412	284	28	p	p	NOUN
ejpam-4412	284	29	-	-	PUNCT
ejpam-4412	284	30	wa(s	wa(s	NUM
ejpam-4412	284	31	,	,	PUNCT
ejpam-4412	284	32	t	t	NOUN
ejpam-4412	284	33	)	)	PUNCT
ejpam-4412	284	34	i.e.	i.e.	X
ejpam-4412	284	35	,	,	PUNCT
ejpam-4412	284	36	t2	t2	PROPN
ejpam-4412	284	37	has	have	VERB
ejpam-4412	284	38	no	no	DET
ejpam-4412	284	39	non	non	ADJ
ejpam-4412	284	40	-	-	ADJ
ejpam-4412	284	41	zero	zero	ADJ
ejpam-4412	284	42	invariant	invariant	ADJ
ejpam-4412	284	43	subspace	subspace	NOUN
ejpam-4412	284	44	m	m	VERB
ejpam-4412	284	45	such	such	ADJ
ejpam-4412	284	46	that	that	SCONJ
ejpam-4412	284	47	t2|m	t2|m	NOUN
ejpam-4412	284	48	is	be	AUX
ejpam-4412	284	49	normal	normal	ADJ
ejpam-4412	284	50	.	.	PUNCT
ejpam-4412	285	1	lemma	lemma	PROPN
ejpam-4412	285	2	9	9	NUM
ejpam-4412	285	3	.	.	PUNCT
ejpam-4412	286	1	let	let	VERB
ejpam-4412	286	2	0	0	NUM
ejpam-4412	286	3	<	<	X
ejpam-4412	286	4	s	s	PROPN
ejpam-4412	286	5	,	,	PUNCT
ejpam-4412	286	6	t	t	PROPN
ejpam-4412	286	7	,	,	PUNCT
ejpam-4412	286	8	s+	s+	ADP
ejpam-4412	286	9	t	t	PROPN
ejpam-4412	286	10	=	=	SYM
ejpam-4412	286	11	1	1	NUM
ejpam-4412	286	12	and	and	CCONJ
ejpam-4412	286	13	0	0	NUM
ejpam-4412	286	14	<	<	X
ejpam-4412	286	15	p	p	X
ejpam-4412	286	16	≤	≤	NUM
ejpam-4412	286	17	1	1	NUM
ejpam-4412	286	18	.	.	PUNCT
ejpam-4412	287	1	let	let	AUX
ejpam-4412	287	2	t	t	NOUN
ejpam-4412	287	3	=	=	SYM
ejpam-4412	287	4	u	u	SYM
ejpam-4412	287	5	|t	|t	VERB
ejpam-4412	287	6	|	|	PROPN
ejpam-4412	287	7	∈	∈	PROPN
ejpam-4412	287	8	b(h	b(h	PROPN
ejpam-4412	287	9	)	)	PUNCT
ejpam-4412	287	10	be	be	AUX
ejpam-4412	287	11	class	class	NOUN
ejpam-4412	287	12	p	p	NOUN
ejpam-4412	287	13	-	-	PUNCT
ejpam-4412	287	14	wa(s	wa(s	NUM
ejpam-4412	287	15	,	,	PUNCT
ejpam-4412	287	16	t	t	NOUN
ejpam-4412	287	17	)	)	PUNCT
ejpam-4412	287	18	and	and	CCONJ
ejpam-4412	287	19	ker(t	ker(t	NOUN
ejpam-4412	287	20	)	)	PUNCT
ejpam-4412	288	1	⊂	⊂	PROPN
ejpam-4412	288	2	ker(t	ker(t	NOUN
ejpam-4412	288	3	∗	∗	NOUN
ejpam-4412	288	4	)	)	PUNCT
ejpam-4412	288	5	.	.	PUNCT
ejpam-4412	289	1	suppose	suppose	VERB
ejpam-4412	290	1	t	t	PROPN
ejpam-4412	290	2	(	(	PUNCT
ejpam-4412	290	3	s	s	PROPN
ejpam-4412	290	4	,	,	PUNCT
ejpam-4412	290	5	t	t	PROPN
ejpam-4412	290	6	)	)	PUNCT
ejpam-4412	290	7	=	=	PUNCT
ejpam-4412	290	8	|t	|t	PROPN
ejpam-4412	291	1	|su	|su	NOUN
ejpam-4412	291	2	|t	|t	VERB
ejpam-4412	291	3	|t	|t	VERB
ejpam-4412	291	4	be	be	AUX
ejpam-4412	291	5	of	of	ADP
ejpam-4412	291	6	the	the	DET
ejpam-4412	291	7	form	form	NOUN
ejpam-4412	291	8	n⊕t	n⊕t	NOUN
ejpam-4412	291	9	′	′	NOUN
ejpam-4412	291	10	on	on	ADP
ejpam-4412	291	11	h	h	NOUN
ejpam-4412	291	12	=	=	NOUN
ejpam-4412	291	13	m⊕m⊥	m⊕m⊥	NOUN
ejpam-4412	291	14	,	,	PUNCT
ejpam-4412	291	15	where	where	SCONJ
ejpam-4412	291	16	n	n	PRON
ejpam-4412	291	17	is	be	AUX
ejpam-4412	291	18	a	a	DET
ejpam-4412	291	19	normal	normal	ADJ
ejpam-4412	291	20	operator	operator	NOUN
ejpam-4412	291	21	on	on	ADP
ejpam-4412	291	22	m.	m.	NOUN
ejpam-4412	291	23	then	then	ADV
ejpam-4412	291	24	t	t	PROPN
ejpam-4412	291	25	=	=	SYM
ejpam-4412	291	26	n	n	PROPN
ejpam-4412	291	27	⊕	⊕	PROPN
ejpam-4412	291	28	t1	t1	NOUN
ejpam-4412	291	29	and	and	CCONJ
ejpam-4412	291	30	u	u	X
ejpam-4412	291	31	=	=	PROPN
ejpam-4412	291	32	u11	u11	PROPN
ejpam-4412	291	33	⊕	⊕	PROPN
ejpam-4412	291	34	u22	u22	PROPN
ejpam-4412	291	35	where	where	SCONJ
ejpam-4412	291	36	t1	t1	PROPN
ejpam-4412	291	37	is	be	AUX
ejpam-4412	291	38	class	class	NOUN
ejpam-4412	291	39	p	p	NOUN
ejpam-4412	291	40	-	-	PUNCT
ejpam-4412	291	41	wa(s	wa(s	NUM
ejpam-4412	291	42	,	,	PUNCT
ejpam-4412	291	43	t	t	PROPN
ejpam-4412	291	44	)	)	PUNCT
ejpam-4412	291	45	with	with	ADP
ejpam-4412	291	46	ker(t1	ker(t1	PROPN
ejpam-4412	291	47	)	)	PUNCT
ejpam-4412	292	1	⊂	⊂	PROPN
ejpam-4412	292	2	ker(t	ker(t	PROPN
ejpam-4412	292	3	∗	∗	NOUN
ejpam-4412	292	4	1	1	NUM
ejpam-4412	292	5	)	)	PUNCT
ejpam-4412	292	6	and	and	CCONJ
ejpam-4412	292	7	n	n	CCONJ
ejpam-4412	292	8	=	=	PRON
ejpam-4412	292	9	u11|n	u11|n	NOUN
ejpam-4412	293	1	|	|	ADV
ejpam-4412	293	2	is	be	AUX
ejpam-4412	293	3	the	the	DET
ejpam-4412	293	4	polar	polar	ADJ
ejpam-4412	293	5	decomposition	decomposition	NOUN
ejpam-4412	293	6	of	of	ADP
ejpam-4412	293	7	n	n	PROPN
ejpam-4412	293	8	.	.	PUNCT
ejpam-4412	294	1	proof	proof	NOUN
ejpam-4412	294	2	.	.	PUNCT
ejpam-4412	295	1	since	since	SCONJ
ejpam-4412	295	2	|t	|t	PROPN
ejpam-4412	295	3	(	(	PUNCT
ejpam-4412	295	4	s	s	PROPN
ejpam-4412	295	5	,	,	PUNCT
ejpam-4412	295	6	t)|2rp	t)|2rp	ADJ
ejpam-4412	295	7	≥	≥	NOUN
ejpam-4412	295	8	|t	|t	ADV
ejpam-4412	295	9	|2rp	|2rp	ADJ
ejpam-4412	295	10	≥	≥	NUM
ejpam-4412	295	11	|(t	|(t	PROPN
ejpam-4412	295	12	(	(	PUNCT
ejpam-4412	295	13	s	s	PROPN
ejpam-4412	295	14	,	,	PUNCT
ejpam-4412	295	15	t))∗|2rp	t))∗|2rp	VERB
ejpam-4412	295	16	for	for	ADP
ejpam-4412	295	17	r	r	NOUN
ejpam-4412	295	18	∈	∈	PROPN
ejpam-4412	295	19	min{s	min{s	PROPN
ejpam-4412	295	20	,	,	PUNCT
ejpam-4412	295	21	t	t	PROPN
ejpam-4412	295	22	}	}	PUNCT
ejpam-4412	295	23	,	,	PUNCT
ejpam-4412	295	24	we	we	PRON
ejpam-4412	295	25	have	have	VERB
ejpam-4412	295	26	|n	|n	ADJ
ejpam-4412	295	27	|2rp	|2rp	PROPN
ejpam-4412	295	28	⊕	⊕	PROPN
ejpam-4412	295	29	|t	|t	VERB
ejpam-4412	295	30	′|2rp	′|2rp	VERB
ejpam-4412	295	31	≥	≥	NOUN
ejpam-4412	295	32	|t	|t	ADV
ejpam-4412	296	1	|2rp	|2rp	PROPN
ejpam-4412	296	2	≥	≥	NUM
ejpam-4412	296	3	|n	|n	NOUN
ejpam-4412	296	4	|2rp	|2rp	PROPN
ejpam-4412	296	5	⊕	⊕	PROPN
ejpam-4412	296	6	|t	|t	VERB
ejpam-4412	296	7	′∗|2rp	′∗|2rp	ADJ
ejpam-4412	296	8	by	by	ADP
ejpam-4412	296	9	assumption	assumption	NOUN
ejpam-4412	296	10	.	.	PUNCT
ejpam-4412	297	1	this	this	PRON
ejpam-4412	297	2	implies	imply	VERB
ejpam-4412	297	3	that	that	PRON
ejpam-4412	297	4	|t	|t	PROPN
ejpam-4412	298	1	|	|	INTJ
ejpam-4412	298	2	is	be	AUX
ejpam-4412	298	3	of	of	ADP
ejpam-4412	298	4	the	the	DET
ejpam-4412	298	5	form	form	NOUN
ejpam-4412	298	6	|n	|n	NOUN
ejpam-4412	298	7	|	|	ADV
ejpam-4412	298	8	⊕l	⊕l	NOUN
ejpam-4412	298	9	for	for	ADP
ejpam-4412	298	10	some	some	DET
ejpam-4412	298	11	positive	positive	ADJ
ejpam-4412	298	12	operator	operator	NOUN
ejpam-4412	298	13	l.	l.	NOUN
ejpam-4412	298	14	let	let	VERB
ejpam-4412	298	15	u	u	PRON
ejpam-4412	298	16	=	=	PUNCT
ejpam-4412	298	17	(	(	PUNCT
ejpam-4412	298	18	u11	u11	PROPN
ejpam-4412	298	19	u12	u12	PROPN
ejpam-4412	298	20	u21	u21	PROPN
ejpam-4412	298	21	u22	u22	PROPN
ejpam-4412	298	22	)	)	PUNCT
ejpam-4412	298	23	be	be	VERB
ejpam-4412	298	24	2×2	2×2	NUM
ejpam-4412	298	25	matrix	matrix	NOUN
ejpam-4412	298	26	representation	representation	NOUN
ejpam-4412	298	27	of	of	ADP
ejpam-4412	298	28	u	u	NOUN
ejpam-4412	298	29	with	with	ADP
ejpam-4412	298	30	respect	respect	NOUN
ejpam-4412	298	31	to	to	ADP
ejpam-4412	298	32	the	the	DET
ejpam-4412	298	33	decomposition	decomposition	NOUN
ejpam-4412	298	34	h	h	NOUN
ejpam-4412	299	1	=	=	PUNCT
ejpam-4412	299	2	m⊕m⊥.	m⊕m⊥.	NOUN
ejpam-4412	300	1	then	then	ADV
ejpam-4412	300	2	the	the	DET
ejpam-4412	300	3	definition	definition	NOUN
ejpam-4412	300	4	t	t	PROPN
ejpam-4412	300	5	(	(	PUNCT
ejpam-4412	300	6	s	s	PROPN
ejpam-4412	300	7	,	,	PUNCT
ejpam-4412	300	8	t	t	PROPN
ejpam-4412	300	9	)	)	PUNCT
ejpam-4412	300	10	means	mean	NOUN
ejpam-4412	300	11	(	(	PUNCT
ejpam-4412	300	12	n	n	ADV
ejpam-4412	300	13	0	0	NUM
ejpam-4412	300	14	0	0	NUM
ejpam-4412	300	15	t	t	NOUN
ejpam-4412	300	16	′	′	NUM
ejpam-4412	300	17	)	)	PUNCT
ejpam-4412	301	1	=	=	PRON
ejpam-4412	301	2	(	(	PUNCT
ejpam-4412	301	3	|n	|n	X
ejpam-4412	301	4	|s	|s	PROPN
ejpam-4412	301	5	0	0	NUM
ejpam-4412	301	6	0	0	NUM
ejpam-4412	301	7	ls	ls	NOUN
ejpam-4412	301	8	)	)	PUNCT
ejpam-4412	301	9	(	(	PUNCT
ejpam-4412	301	10	u11	u11	PROPN
ejpam-4412	301	11	u12	u12	PROPN
ejpam-4412	301	12	u21	u21	PROPN
ejpam-4412	301	13	u22	u22	PROPN
ejpam-4412	301	14	)	)	PUNCT
ejpam-4412	301	15	(	(	PUNCT
ejpam-4412	301	16	|n	|n	X
ejpam-4412	301	17	|t	|t	PROPN
ejpam-4412	301	18	0	0	NUM
ejpam-4412	301	19	0	0	NUM
ejpam-4412	302	1	lt	lt	PRON
ejpam-4412	302	2	)	)	PUNCT
ejpam-4412	302	3	hence	hence	ADV
ejpam-4412	302	4	,	,	PUNCT
ejpam-4412	302	5	we	we	PRON
ejpam-4412	302	6	have	have	VERB
ejpam-4412	302	7	n	n	NOUN
ejpam-4412	302	8	=	=	X
ejpam-4412	302	9	|n	|n	X
ejpam-4412	302	10	|su11|n	|su11|n	VERB
ejpam-4412	302	11	|t	|t	PROPN
ejpam-4412	302	12	,	,	PUNCT
ejpam-4412	302	13	|n	|n	NOUN
ejpam-4412	302	14	|su12l	|su12l	PROPN
ejpam-4412	302	15	t	t	PROPN
ejpam-4412	303	1	=	=	PUNCT
ejpam-4412	303	2	0	0	NUM
ejpam-4412	303	3	and	and	CCONJ
ejpam-4412	303	4	lsu21|n	lsu21|n	PROPN
ejpam-4412	303	5	|t	|t	PROPN
ejpam-4412	304	1	=	=	NOUN
ejpam-4412	304	2	0	0	X
ejpam-4412	304	3	.	.	PUNCT
ejpam-4412	305	1	since	since	SCONJ
ejpam-4412	305	2	ker(t	ker(t	NOUN
ejpam-4412	305	3	)	)	PUNCT
ejpam-4412	306	1	⊂	⊂	PROPN
ejpam-4412	306	2	ker(t	ker(t	NOUN
ejpam-4412	306	3	∗	∗	NOUN
ejpam-4412	306	4	)	)	PUNCT
ejpam-4412	306	5	,	,	PUNCT
ejpam-4412	306	6	ran(u	ran(u	NOUN
ejpam-4412	306	7	)	)	PUNCT
ejpam-4412	306	8	=	=	SYM
ejpam-4412	306	9	ran(t	ran(t	NOUN
ejpam-4412	306	10	)	)	PUNCT
ejpam-4412	307	1	=	=	PUNCT
ejpam-4412	307	2	ker(t	ker(t	NOUN
ejpam-4412	307	3	∗)⊥	∗)⊥	NUM
ejpam-4412	307	4	⊂	⊂	NOUN
ejpam-4412	307	5	ker(t	ker(t	NOUN
ejpam-4412	307	6	)	)	PUNCT
ejpam-4412	307	7	⊥	⊥	NOUN
ejpam-4412	307	8	=	=	PUNCT
ejpam-4412	308	1	ran(|t	ran(|t	ADP
ejpam-4412	308	2	|	|	ADV
ejpam-4412	308	3	)	)	PUNCT
ejpam-4412	308	4	.	.	PUNCT
ejpam-4412	309	1	let	let	VERB
ejpam-4412	309	2	nx	nx	X
ejpam-4412	309	3	=	=	SYM
ejpam-4412	309	4	0	0	NUM
ejpam-4412	309	5	for	for	ADP
ejpam-4412	309	6	x	x	PROPN
ejpam-4412	309	7	∈	∈	PROPN
ejpam-4412	309	8	m.	m.	NOUN
ejpam-4412	309	9	then	then	ADV
ejpam-4412	309	10	x	x	SYM
ejpam-4412	309	11	∈	∈	PROPN
ejpam-4412	309	12	ker(|t	ker(|t	X
ejpam-4412	309	13	|	|	ADV
ejpam-4412	309	14	)	)	PUNCT
ejpam-4412	310	1	=	=	SYM
ejpam-4412	310	2	ker(u	ker(u	PROPN
ejpam-4412	310	3	)	)	PUNCT
ejpam-4412	310	4	,	,	PUNCT
ejpam-4412	310	5	and	and	CCONJ
ejpam-4412	310	6	ux	ux	NOUN
ejpam-4412	311	1	=	=	SYM
ejpam-4412	311	2	(	(	PUNCT
ejpam-4412	311	3	u11	u11	PROPN
ejpam-4412	311	4	u12	u12	PROPN
ejpam-4412	311	5	u21	u21	PROPN
ejpam-4412	311	6	u22	u22	PROPN
ejpam-4412	311	7	)	)	PUNCT
ejpam-4412	311	8	(	(	PUNCT
ejpam-4412	311	9	x	x	SYM
ejpam-4412	311	10	0	0	NUM
ejpam-4412	311	11	)	)	PUNCT
ejpam-4412	311	12	=	=	SYM
ejpam-4412	311	13	(	(	PUNCT
ejpam-4412	311	14	u11x	u11x	NOUN
ejpam-4412	311	15	u21x	u21x	X
ejpam-4412	311	16	)	)	PUNCT
ejpam-4412	312	1	=	=	PUNCT
ejpam-4412	312	2	0	0	X
ejpam-4412	312	3	.	.	PUNCT
ejpam-4412	313	1	hence	hence	ADV
ejpam-4412	313	2	ker(n	ker(n	PROPN
ejpam-4412	313	3	)	)	PUNCT
ejpam-4412	313	4	⊂	⊂	PROPN
ejpam-4412	313	5	ker(u11	ker(u11	PROPN
ejpam-4412	313	6	)	)	PUNCT
ejpam-4412	313	7	∩	∩	PROPN
ejpam-4412	313	8	ker(u21	ker(u21	NOUN
ejpam-4412	313	9	)	)	PUNCT
ejpam-4412	313	10	.	.	PUNCT
ejpam-4412	314	1	let	let	VERB
ejpam-4412	314	2	x	x	X
ejpam-4412	314	3	∈	∈	PROPN
ejpam-4412	314	4	m.	m.	NOUN
ejpam-4412	314	5	then	then	ADV
ejpam-4412	314	6	u	u	X
ejpam-4412	314	7	(	(	PUNCT
ejpam-4412	314	8	x	x	NOUN
ejpam-4412	314	9	0	0	NUM
ejpam-4412	314	10	)	)	PUNCT
ejpam-4412	314	11	=	=	SYM
ejpam-4412	314	12	(	(	PUNCT
ejpam-4412	315	1	u11x	u11x	X
ejpam-4412	315	2	u21x	u21x	SYM
ejpam-4412	315	3	)	)	PUNCT
ejpam-4412	315	4	∈	∈	PROPN
ejpam-4412	315	5	ran(|t	ran(|t	NOUN
ejpam-4412	315	6	|	|	ADV
ejpam-4412	315	7	)	)	PUNCT
ejpam-4412	316	1	=	=	VERB
ejpam-4412	316	2	ran(|n	ran(|n	VERB
ejpam-4412	316	3	|	|	PROPN
ejpam-4412	316	4	⊕	⊕	PROPN
ejpam-4412	316	5	l	l	PROPN
ejpam-4412	316	6	)	)	PUNCT
ejpam-4412	316	7	.	.	PUNCT
ejpam-4412	317	1	hence	hence	ADV
ejpam-4412	317	2	ran(u11	ran(u11	VERB
ejpam-4412	317	3	)	)	PUNCT
ejpam-4412	317	4	⊂	⊂	PRON
ejpam-4412	317	5	ran(|n	ran(|n	VERB
ejpam-4412	317	6	|	|	ADV
ejpam-4412	317	7	)	)	PUNCT
ejpam-4412	317	8	,	,	PUNCT
ejpam-4412	317	9	ran(u21	ran(u21	NOUN
ejpam-4412	317	10	)	)	PUNCT
ejpam-4412	317	11	⊂	⊂	PROPN
ejpam-4412	317	12	ran(l	ran(l	PROPN
ejpam-4412	317	13	)	)	PUNCT
ejpam-4412	317	14	.	.	PUNCT
ejpam-4412	318	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	318	2	,	,	PUNCT
ejpam-4412	318	3	n.	n.	NOUN
ejpam-4412	318	4	h.	h.	PROPN
ejpam-4412	318	5	altaweel	altaweel	PROPN
ejpam-4412	318	6	/	/	SYM
ejpam-4412	318	7	eur	eur	PROPN
ejpam-4412	318	8	.	.	PUNCT
ejpam-4412	319	1	j.	j.	PROPN
ejpam-4412	319	2	pure	pure	PROPN
ejpam-4412	319	3	appl	appl	PROPN
ejpam-4412	319	4	.	.	PROPN
ejpam-4412	319	5	math	math	PROPN
ejpam-4412	319	6	,	,	PUNCT
ejpam-4412	319	7	15	15	NUM
ejpam-4412	319	8	(	(	PUNCT
ejpam-4412	319	9	3	3	NUM
ejpam-4412	319	10	)	)	PUNCT
ejpam-4412	319	11	(	(	PUNCT
ejpam-4412	319	12	2022	2022	NUM
ejpam-4412	319	13	)	)	PUNCT
ejpam-4412	319	14	,	,	PUNCT
ejpam-4412	319	15	1067	1067	NUM
ejpam-4412	319	16	-	-	SYM
ejpam-4412	319	17	1089	1089	NUM
ejpam-4412	319	18	1078	1078	NUM
ejpam-4412	319	19	similarly	similarly	ADV
ejpam-4412	319	20	ran(u12	ran(u12	NOUN
ejpam-4412	319	21	)	)	PUNCT
ejpam-4412	320	1	⊂	⊂	PRON
ejpam-4412	320	2	ran(|n	ran(|n	VERB
ejpam-4412	320	3	|	|	ADV
ejpam-4412	320	4	)	)	PUNCT
ejpam-4412	320	5	,	,	PUNCT
ejpam-4412	320	6	ran(u22	ran(u22	INTJ
ejpam-4412	320	7	)	)	PUNCT
ejpam-4412	321	1	⊂	⊂	PROPN
ejpam-4412	321	2	ran(l	ran(l	PROPN
ejpam-4412	321	3	)	)	PUNCT
ejpam-4412	321	4	.	.	PUNCT
ejpam-4412	322	1	let	let	VERB
ejpam-4412	322	2	lx	lx	VERB
ejpam-4412	322	3	=	=	NOUN
ejpam-4412	322	4	0	0	NUM
ejpam-4412	322	5	for	for	ADP
ejpam-4412	322	6	x	x	PROPN
ejpam-4412	322	7	∈	∈	PROPN
ejpam-4412	322	8	m⊥.	m⊥.	ADJ
ejpam-4412	322	9	then	then	ADV
ejpam-4412	323	1	x	x	SYM
ejpam-4412	323	2	∈	∈	PROPN
ejpam-4412	323	3	ker(|t	ker(|t	X
ejpam-4412	323	4	|	|	ADV
ejpam-4412	323	5	)	)	PUNCT
ejpam-4412	323	6	=	=	SYM
ejpam-4412	323	7	ker(u	ker(u	PROPN
ejpam-4412	323	8	)	)	PUNCT
ejpam-4412	323	9	and	and	CCONJ
ejpam-4412	323	10	u	u	X
ejpam-4412	323	11	(	(	PUNCT
ejpam-4412	323	12	0	0	NUM
ejpam-4412	323	13	x	x	X
ejpam-4412	323	14	)	)	PUNCT
ejpam-4412	323	15	=	=	SYM
ejpam-4412	324	1	(	(	PUNCT
ejpam-4412	324	2	u12x	u12x	NOUN
ejpam-4412	324	3	u22x	u22x	PROPN
ejpam-4412	324	4	)	)	PUNCT
ejpam-4412	324	5	=	=	SYM
ejpam-4412	325	1	0	0	PUNCT
ejpam-4412	325	2	hence	hence	ADV
ejpam-4412	325	3	ker(l	ker(l	PROPN
ejpam-4412	325	4	)	)	PUNCT
ejpam-4412	325	5	⊂	⊂	PROPN
ejpam-4412	325	6	ker(u12	ker(u12	NOUN
ejpam-4412	325	7	)	)	PUNCT
ejpam-4412	325	8	∩	∩	NOUN
ejpam-4412	325	9	ker(u22	ker(u22	NOUN
ejpam-4412	325	10	)	)	PUNCT
ejpam-4412	325	11	.	.	PUNCT
ejpam-4412	326	1	let	let	VERB
ejpam-4412	326	2	n	n	NOUN
ejpam-4412	326	3	=	=	PUNCT
ejpam-4412	326	4	v	v	PROPN
ejpam-4412	326	5	|n	|n	NOUN
ejpam-4412	326	6	|	|	ADV
ejpam-4412	326	7	be	be	AUX
ejpam-4412	326	8	the	the	DET
ejpam-4412	326	9	polar	polar	ADJ
ejpam-4412	326	10	decomposition	decomposition	NOUN
ejpam-4412	326	11	of	of	ADP
ejpam-4412	326	12	n	n	PROPN
ejpam-4412	326	13	.	.	PUNCT
ejpam-4412	327	1	then	then	ADV
ejpam-4412	327	2	(	(	PUNCT
ejpam-4412	327	3	v	v	X
ejpam-4412	327	4	|n	|n	X
ejpam-4412	327	5	|s	|s	PROPN
ejpam-4412	327	6	−	−	PROPN
ejpam-4412	327	7	|n	|n	X
ejpam-4412	327	8	|su11)|n	|su11)|n	PROPN
ejpam-4412	327	9	|t	|t	VERB
ejpam-4412	327	10	=	=	NOUN
ejpam-4412	327	11	0	0	X
ejpam-4412	327	12	.	.	PUNCT
ejpam-4412	328	1	hence	hence	ADV
ejpam-4412	328	2	v	v	PROPN
ejpam-4412	328	3	|n	|n	X
ejpam-4412	328	4	|s	|s	PROPN
ejpam-4412	328	5	−	−	PROPN
ejpam-4412	328	6	|n	|n	AUX
ejpam-4412	328	7	|su11	|su11	VERB
ejpam-4412	328	8	=	=	SYM
ejpam-4412	328	9	0	0	NUM
ejpam-4412	328	10	on	on	ADP
ejpam-4412	328	11	ran(|n	ran(|n	NOUN
ejpam-4412	328	12	|	|	ADV
ejpam-4412	328	13	)	)	PUNCT
ejpam-4412	328	14	.	.	PUNCT
ejpam-4412	329	1	since	since	SCONJ
ejpam-4412	329	2	ker(n	ker(n	X
ejpam-4412	329	3	)	)	PUNCT
ejpam-4412	329	4	⊂	⊂	PROPN
ejpam-4412	329	5	ker(u11	ker(u11	PROPN
ejpam-4412	329	6	)	)	PUNCT
ejpam-4412	329	7	,	,	PUNCT
ejpam-4412	329	8	this	this	PRON
ejpam-4412	329	9	implies	imply	VERB
ejpam-4412	329	10	0	0	NUM
ejpam-4412	329	11	=	=	SYM
ejpam-4412	329	12	v	v	PROPN
ejpam-4412	329	13	|n	|n	X
ejpam-4412	329	14	|s	|s	PROPN
ejpam-4412	329	15	−	−	PROPN
ejpam-4412	329	16	|n	|n	AUX
ejpam-4412	329	17	|su11	|su11	VERB
ejpam-4412	329	18	=	=	SYM
ejpam-4412	329	19	|n	|n	X
ejpam-4412	329	20	|s(v	|s(v	PROPN
ejpam-4412	329	21	−	−	PROPN
ejpam-4412	329	22	u11	u11	PROPN
ejpam-4412	329	23	)	)	PUNCT
ejpam-4412	329	24	.	.	PUNCT
ejpam-4412	330	1	hence	hence	ADV
ejpam-4412	330	2	ran(v	ran(v	VERB
ejpam-4412	330	3	−	−	PROPN
ejpam-4412	330	4	u11	u11	PROPN
ejpam-4412	330	5	)	)	PUNCT
ejpam-4412	330	6	⊂	⊂	PROPN
ejpam-4412	330	7	ker(|n	ker(|n	PROPN
ejpam-4412	330	8	|	|	ADV
ejpam-4412	330	9	)	)	PUNCT
ejpam-4412	330	10	∩	∩	NOUN
ejpam-4412	330	11	ran(|n	ran(|n	VERB
ejpam-4412	330	12	|	|	NOUN
ejpam-4412	330	13	)	)	PUNCT
ejpam-4412	330	14	=	=	PUNCT
ejpam-4412	330	15	{	{	PUNCT
ejpam-4412	330	16	0	0	NUM
ejpam-4412	330	17	}	}	PUNCT
ejpam-4412	330	18	.	.	PUNCT
ejpam-4412	331	1	hence	hence	ADV
ejpam-4412	331	2	v	v	NOUN
ejpam-4412	331	3	=	=	SYM
ejpam-4412	331	4	u11	u11	ADJ
ejpam-4412	331	5	and	and	CCONJ
ejpam-4412	331	6	n	n	CCONJ
ejpam-4412	331	7	=	=	PRON
ejpam-4412	331	8	u11|n	u11|n	NOUN
ejpam-4412	332	1	|	|	ADV
ejpam-4412	332	2	is	be	AUX
ejpam-4412	332	3	the	the	DET
ejpam-4412	332	4	polar	polar	ADJ
ejpam-4412	332	5	decomposition	decomposition	NOUN
ejpam-4412	332	6	of	of	ADP
ejpam-4412	332	7	n	n	PROPN
ejpam-4412	332	8	.	.	PUNCT
ejpam-4412	333	1	since	since	SCONJ
ejpam-4412	333	2	|n	|n	X
ejpam-4412	333	3	|su12l	|su12l	X
ejpam-4412	333	4	t	t	NOUN
ejpam-4412	333	5	=	=	SYM
ejpam-4412	333	6	0	0	NUM
ejpam-4412	333	7	,	,	PUNCT
ejpam-4412	333	8	ran(u11l	ran(u11l	VERB
ejpam-4412	333	9	t	t	PROPN
ejpam-4412	333	10	)	)	PUNCT
ejpam-4412	333	11	⊂	⊂	PROPN
ejpam-4412	333	12	ker(|n	ker(|n	PROPN
ejpam-4412	334	1	|	|	ADV
ejpam-4412	334	2	)	)	PUNCT
ejpam-4412	335	1	∩	∩	NOUN
ejpam-4412	335	2	ran(|n	ran(|n	VERB
ejpam-4412	335	3	|	|	NOUN
ejpam-4412	335	4	)	)	PUNCT
ejpam-4412	335	5	=	=	PUNCT
ejpam-4412	335	6	{	{	PUNCT
ejpam-4412	335	7	0	0	NUM
ejpam-4412	335	8	}	}	PUNCT
ejpam-4412	335	9	.	.	PUNCT
ejpam-4412	336	1	hence	hence	ADV
ejpam-4412	336	2	u12l	u12l	PROPN
ejpam-4412	336	3	t	t	PROPN
ejpam-4412	336	4	and	and	CCONJ
ejpam-4412	336	5	u12	u12	PROPN
ejpam-4412	336	6	=	=	SYM
ejpam-4412	336	7	0	0	X
ejpam-4412	336	8	.	.	PUNCT
ejpam-4412	337	1	similarly	similarly	ADV
ejpam-4412	337	2	we	we	PRON
ejpam-4412	337	3	have	have	VERB
ejpam-4412	337	4	u21	u21	NOUN
ejpam-4412	337	5	=	=	SYM
ejpam-4412	337	6	0	0	NUM
ejpam-4412	337	7	by	by	ADP
ejpam-4412	337	8	lsu21|n	lsu21|n	PROPN
ejpam-4412	337	9	|t	|t	PROPN
ejpam-4412	338	1	=	=	NOUN
ejpam-4412	338	2	0	0	X
ejpam-4412	338	3	.	.	PUNCT
ejpam-4412	339	1	hence	hence	ADV
ejpam-4412	339	2	u	u	NOUN
ejpam-4412	339	3	=	=	SYM
ejpam-4412	339	4	u11	u11	PROPN
ejpam-4412	339	5	⊕	⊕	PROPN
ejpam-4412	339	6	u22	u22	PROPN
ejpam-4412	339	7	.	.	PUNCT
ejpam-4412	340	1	so	so	ADV
ejpam-4412	340	2	we	we	PRON
ejpam-4412	340	3	obtain	obtain	VERB
ejpam-4412	340	4	t	t	NOUN
ejpam-4412	340	5	=	=	SYM
ejpam-4412	340	6	u	u	NOUN
ejpam-4412	340	7	|t	|t	NOUN
ejpam-4412	340	8	|	|	ADV
ejpam-4412	340	9	=	=	SYM
ejpam-4412	340	10	u11|n	u11|n	PROPN
ejpam-4412	341	1	|	|	PROPN
ejpam-4412	341	2	⊕	⊕	PROPN
ejpam-4412	341	3	u22l	u22l	NOUN
ejpam-4412	342	1	=	=	SYM
ejpam-4412	342	2	n	n	NUM
ejpam-4412	342	3	⊕	⊕	PROPN
ejpam-4412	342	4	t1	t1	PROPN
ejpam-4412	342	5	,	,	PUNCT
ejpam-4412	342	6	where	where	SCONJ
ejpam-4412	342	7	t1	t1	NOUN
ejpam-4412	342	8	=	=	SYM
ejpam-4412	342	9	u22l	u22l	PROPN
ejpam-4412	342	10	.	.	PUNCT
ejpam-4412	343	1	3	3	X
ejpam-4412	343	2	.	.	X
ejpam-4412	343	3	quasisimilarity	quasisimilarity	NOUN
ejpam-4412	343	4	an	an	DET
ejpam-4412	343	5	operator	operator	NOUN
ejpam-4412	343	6	x	x	SYM
ejpam-4412	343	7	∈	∈	PROPN
ejpam-4412	343	8	b(k	b(k	PROPN
ejpam-4412	343	9	,	,	PUNCT
ejpam-4412	343	10	h	h	NOUN
ejpam-4412	343	11	)	)	PUNCT
ejpam-4412	343	12	is	be	AUX
ejpam-4412	343	13	called	call	VERB
ejpam-4412	343	14	quasiaffinity	quasiaffinity	NOUN
ejpam-4412	343	15	if	if	SCONJ
ejpam-4412	343	16	x	x	PRON
ejpam-4412	343	17	is	be	AUX
ejpam-4412	343	18	both	both	PRON
ejpam-4412	343	19	injective	injective	ADJ
ejpam-4412	343	20	and	and	CCONJ
ejpam-4412	343	21	has	have	VERB
ejpam-4412	343	22	a	a	DET
ejpam-4412	343	23	dense	dense	ADJ
ejpam-4412	343	24	range	range	NOUN
ejpam-4412	343	25	.	.	PUNCT
ejpam-4412	344	1	for	for	ADP
ejpam-4412	344	2	t	t	PROPN
ejpam-4412	344	3	∈	∈	PROPN
ejpam-4412	344	4	b(h	b(h	PROPN
ejpam-4412	344	5	)	)	PUNCT
ejpam-4412	344	6	and	and	CCONJ
ejpam-4412	344	7	s	s	PROPN
ejpam-4412	344	8	∈	∈	PROPN
ejpam-4412	344	9	b(k	b(k	PROPN
ejpam-4412	344	10	)	)	PUNCT
ejpam-4412	344	11	,	,	PUNCT
ejpam-4412	344	12	if	if	SCONJ
ejpam-4412	344	13	there	there	PRON
ejpam-4412	344	14	exist	exist	VERB
ejpam-4412	344	15	quasiaffinities	quasiaffinitie	NOUN
ejpam-4412	344	16	x	x	SYM
ejpam-4412	344	17	∈	∈	PROPN
ejpam-4412	344	18	b(k	b(k	PROPN
ejpam-4412	344	19	,	,	PUNCT
ejpam-4412	344	20	h	h	NOUN
ejpam-4412	344	21	)	)	PUNCT
ejpam-4412	344	22	and	and	CCONJ
ejpam-4412	344	23	y	y	PROPN
ejpam-4412	344	24	∈	∈	PROPN
ejpam-4412	344	25	b(h	b(h	PROPN
ejpam-4412	344	26	,	,	PUNCT
ejpam-4412	344	27	k	k	NOUN
ejpam-4412	344	28	)	)	PUNCT
ejpam-4412	344	29	such	such	ADJ
ejpam-4412	344	30	that	that	SCONJ
ejpam-4412	344	31	tx	tx	PROPN
ejpam-4412	344	32	=	=	SYM
ejpam-4412	344	33	xs	xs	PROPN
ejpam-4412	344	34	and	and	CCONJ
ejpam-4412	344	35	y	y	PROPN
ejpam-4412	344	36	t	t	PROPN
ejpam-4412	344	37	=	=	SYM
ejpam-4412	344	38	sy	sy	PROPN
ejpam-4412	344	39	,	,	PUNCT
ejpam-4412	344	40	then	then	ADV
ejpam-4412	344	41	we	we	PRON
ejpam-4412	344	42	say	say	VERB
ejpam-4412	344	43	that	that	SCONJ
ejpam-4412	344	44	t	t	PROPN
ejpam-4412	344	45	and	and	CCONJ
ejpam-4412	344	46	s	s	VERB
ejpam-4412	344	47	are	be	AUX
ejpam-4412	344	48	quasisimilar	quasisimilar	ADJ
ejpam-4412	344	49	.	.	PUNCT
ejpam-4412	345	1	the	the	DET
ejpam-4412	345	2	operator	operator	NOUN
ejpam-4412	345	3	t	t	PROPN
ejpam-4412	345	4	∈	∈	PROPN
ejpam-4412	345	5	b(h	b(h	PROPN
ejpam-4412	345	6	)	)	PUNCT
ejpam-4412	345	7	is	be	AUX
ejpam-4412	345	8	said	say	VERB
ejpam-4412	345	9	to	to	PART
ejpam-4412	345	10	be	be	AUX
ejpam-4412	345	11	pure	pure	ADJ
ejpam-4412	345	12	if	if	SCONJ
ejpam-4412	345	13	there	there	PRON
ejpam-4412	345	14	exists	exist	VERB
ejpam-4412	345	15	no	no	DET
ejpam-4412	345	16	nontrivial	nontrivial	NOUN
ejpam-4412	345	17	reducing	reduce	VERB
ejpam-4412	345	18	subspace	subspace	NOUN
ejpam-4412	345	19	m	m	NOUN
ejpam-4412	345	20	of	of	ADP
ejpam-4412	345	21	h	h	NOUN
ejpam-4412	345	22	such	such	ADJ
ejpam-4412	345	23	that	that	SCONJ
ejpam-4412	345	24	the	the	DET
ejpam-4412	345	25	restriction	restriction	NOUN
ejpam-4412	345	26	of	of	ADP
ejpam-4412	345	27	t	t	PROPN
ejpam-4412	345	28	to	to	ADP
ejpam-4412	345	29	m	m	PROPN
ejpam-4412	345	30	is	be	AUX
ejpam-4412	345	31	normal	normal	ADJ
ejpam-4412	345	32	and	and	CCONJ
ejpam-4412	345	33	is	be	AUX
ejpam-4412	345	34	completely	completely	ADV
ejpam-4412	345	35	hyponormal	hyponormal	ADJ
ejpam-4412	345	36	if	if	SCONJ
ejpam-4412	345	37	it	it	PRON
ejpam-4412	345	38	is	be	AUX
ejpam-4412	345	39	pure	pure	ADJ
ejpam-4412	345	40	.	.	PUNCT
ejpam-4412	346	1	recall	recall	VERB
ejpam-4412	346	2	that	that	SCONJ
ejpam-4412	346	3	every	every	DET
ejpam-4412	346	4	operator	operator	NOUN
ejpam-4412	346	5	t	t	PROPN
ejpam-4412	346	6	∈	∈	PROPN
ejpam-4412	346	7	b(h	b(h	PROPN
ejpam-4412	346	8	)	)	PUNCT
ejpam-4412	346	9	has	have	VERB
ejpam-4412	346	10	a	a	DET
ejpam-4412	346	11	direct	direct	ADJ
ejpam-4412	346	12	sum	sum	NOUN
ejpam-4412	346	13	decomposition	decomposition	NOUN
ejpam-4412	346	14	t	t	NOUN
ejpam-4412	346	15	=	=	SYM
ejpam-4412	346	16	t1⊕t2	t1⊕t2	PROPN
ejpam-4412	346	17	,	,	PUNCT
ejpam-4412	346	18	where	where	SCONJ
ejpam-4412	346	19	t1	t1	NOUN
ejpam-4412	346	20	and	and	CCONJ
ejpam-4412	346	21	t2	t2	NOUN
ejpam-4412	346	22	are	be	AUX
ejpam-4412	346	23	normal	normal	ADJ
ejpam-4412	346	24	and	and	CCONJ
ejpam-4412	346	25	pure	pure	ADJ
ejpam-4412	346	26	parts	part	NOUN
ejpam-4412	346	27	,	,	PUNCT
ejpam-4412	346	28	respectively	respectively	ADV
ejpam-4412	346	29	.	.	PUNCT
ejpam-4412	347	1	of	of	ADP
ejpam-4412	347	2	course	course	ADV
ejpam-4412	347	3	in	in	ADP
ejpam-4412	347	4	the	the	DET
ejpam-4412	347	5	sum	sum	NOUN
ejpam-4412	347	6	decomposition	decomposition	NOUN
ejpam-4412	347	7	,	,	PUNCT
ejpam-4412	347	8	either	either	CCONJ
ejpam-4412	347	9	t1	t1	NOUN
ejpam-4412	347	10	or	or	CCONJ
ejpam-4412	347	11	t2	t2	NOUN
ejpam-4412	347	12	may	may	AUX
ejpam-4412	347	13	be	be	AUX
ejpam-4412	347	14	absent	absent	ADJ
ejpam-4412	347	15	.	.	PUNCT
ejpam-4412	348	1	the	the	DET
ejpam-4412	348	2	following	follow	VERB
ejpam-4412	348	3	lemma	lemma	PROPN
ejpam-4412	348	4	is	be	AUX
ejpam-4412	348	5	due	due	ADJ
ejpam-4412	348	6	to	to	ADP
ejpam-4412	348	7	williams	williams	PROPN
ejpam-4412	348	8	[	[	X
ejpam-4412	348	9	32	32	NUM
ejpam-4412	348	10	,	,	PUNCT
ejpam-4412	348	11	lemma	lemma	PROPN
ejpam-4412	348	12	1.1	1.1	NUM
ejpam-4412	348	13	]	]	PUNCT
ejpam-4412	348	14	.	.	PUNCT
ejpam-4412	349	1	lemma	lemma	PROPN
ejpam-4412	349	2	10	10	NUM
ejpam-4412	349	3	.	.	PUNCT
ejpam-4412	350	1	let	let	VERB
ejpam-4412	350	2	t	t	PROPN
ejpam-4412	350	3	∈	∈	PROPN
ejpam-4412	350	4	b(h	b(h	PROPN
ejpam-4412	350	5	)	)	PUNCT
ejpam-4412	350	6	and	and	CCONJ
ejpam-4412	350	7	s	s	PROPN
ejpam-4412	350	8	∈	∈	PROPN
ejpam-4412	350	9	b(k	b(k	PROPN
ejpam-4412	350	10	)	)	PUNCT
ejpam-4412	350	11	be	be	AUX
ejpam-4412	350	12	normal	normal	ADJ
ejpam-4412	350	13	operators	operator	NOUN
ejpam-4412	350	14	.	.	PUNCT
ejpam-4412	351	1	it	it	PRON
ejpam-4412	351	2	there	there	ADV
ejpam-4412	351	3	exist	exist	VERB
ejpam-4412	351	4	injective	injective	ADJ
ejpam-4412	351	5	operators	operator	NOUN
ejpam-4412	351	6	x	x	SYM
ejpam-4412	351	7	∈	∈	PROPN
ejpam-4412	351	8	b(k	b(k	PROPN
ejpam-4412	351	9	,	,	PUNCT
ejpam-4412	351	10	h	h	NOUN
ejpam-4412	351	11	)	)	PUNCT
ejpam-4412	351	12	and	and	CCONJ
ejpam-4412	351	13	y	y	PROPN
ejpam-4412	351	14	∈	∈	PROPN
ejpam-4412	351	15	b(h	b(h	PROPN
ejpam-4412	351	16	,	,	PUNCT
ejpam-4412	351	17	k	k	NOUN
ejpam-4412	351	18	)	)	PUNCT
ejpam-4412	351	19	such	such	ADJ
ejpam-4412	351	20	that	that	SCONJ
ejpam-4412	351	21	tx	tx	PROPN
ejpam-4412	351	22	=	=	SYM
ejpam-4412	351	23	xs	xs	PROPN
ejpam-4412	351	24	and	and	CCONJ
ejpam-4412	351	25	y	y	PROPN
ejpam-4412	351	26	t	t	PROPN
ejpam-4412	352	1	=	=	SYM
ejpam-4412	352	2	sy	sy	PROPN
ejpam-4412	352	3	,	,	PUNCT
ejpam-4412	352	4	then	then	ADV
ejpam-4412	352	5	t	t	PROPN
ejpam-4412	352	6	and	and	CCONJ
ejpam-4412	352	7	s	s	PRON
ejpam-4412	352	8	are	be	AUX
ejpam-4412	352	9	unitarily	unitarily	ADV
ejpam-4412	352	10	equivalent	equivalent	ADJ
ejpam-4412	352	11	.	.	PUNCT
ejpam-4412	353	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	353	2	,	,	PUNCT
ejpam-4412	353	3	n.	n.	NOUN
ejpam-4412	353	4	h.	h.	PROPN
ejpam-4412	353	5	altaweel	altaweel	PROPN
ejpam-4412	353	6	/	/	SYM
ejpam-4412	353	7	eur	eur	PROPN
ejpam-4412	353	8	.	.	PUNCT
ejpam-4412	354	1	j.	j.	PROPN
ejpam-4412	354	2	pure	pure	PROPN
ejpam-4412	354	3	appl	appl	PROPN
ejpam-4412	354	4	.	.	PROPN
ejpam-4412	354	5	math	math	PROPN
ejpam-4412	354	6	,	,	PUNCT
ejpam-4412	354	7	15	15	NUM
ejpam-4412	354	8	(	(	PUNCT
ejpam-4412	354	9	3	3	NUM
ejpam-4412	354	10	)	)	PUNCT
ejpam-4412	354	11	(	(	PUNCT
ejpam-4412	354	12	2022	2022	NUM
ejpam-4412	354	13	)	)	PUNCT
ejpam-4412	354	14	,	,	PUNCT
ejpam-4412	354	15	1067	1067	NUM
ejpam-4412	354	16	-	-	SYM
ejpam-4412	354	17	1089	1089	NUM
ejpam-4412	354	18	1079	1079	NUM
ejpam-4412	354	19	corollary	corollary	NOUN
ejpam-4412	354	20	2	2	NUM
ejpam-4412	354	21	.	.	PUNCT
ejpam-4412	355	1	let	let	AUX
ejpam-4412	355	2	t	t	PROPN
ejpam-4412	355	3	∈	∈	PROPN
ejpam-4412	355	4	b(h	b(h	PROPN
ejpam-4412	355	5	)	)	PUNCT
ejpam-4412	355	6	be	be	AUX
ejpam-4412	355	7	class	class	NOUN
ejpam-4412	355	8	p	p	NOUN
ejpam-4412	355	9	-	-	PUNCT
ejpam-4412	355	10	wa(s	wa(s	NUM
ejpam-4412	355	11	,	,	PUNCT
ejpam-4412	355	12	t)operator	t)operator	NOUN
ejpam-4412	355	13	for	for	ADP
ejpam-4412	355	14	0	0	NUM
ejpam-4412	355	15	<	<	X
ejpam-4412	355	16	s	s	PROPN
ejpam-4412	355	17	,	,	PUNCT
ejpam-4412	355	18	t	t	PROPN
ejpam-4412	355	19	,	,	PUNCT
ejpam-4412	355	20	s	s	PART
ejpam-4412	355	21	+	+	NUM
ejpam-4412	355	22	t	t	X
ejpam-4412	355	23	=	=	SYM
ejpam-4412	355	24	1	1	NUM
ejpam-4412	355	25	and	and	CCONJ
ejpam-4412	355	26	0	0	NUM
ejpam-4412	355	27	<	<	X
ejpam-4412	355	28	p	p	X
ejpam-4412	355	29	≤	≤	ADJ
ejpam-4412	355	30	1	1	NUM
ejpam-4412	355	31	.	.	PUNCT
ejpam-4412	356	1	then	then	ADV
ejpam-4412	356	2	t	t	PROPN
ejpam-4412	356	3	=	=	SYM
ejpam-4412	356	4	t1	t1	PROPN
ejpam-4412	356	5	⊕	⊕	PROPN
ejpam-4412	356	6	t2	t2	PROPN
ejpam-4412	356	7	on	on	ADP
ejpam-4412	356	8	the	the	DET
ejpam-4412	356	9	space	space	NOUN
ejpam-4412	356	10	h	h	NOUN
ejpam-4412	356	11	=	=	PRON
ejpam-4412	357	1	h1	h1	PROPN
ejpam-4412	357	2	⊕h2	⊕h2	NOUN
ejpam-4412	357	3	,	,	PUNCT
ejpam-4412	357	4	where	where	SCONJ
ejpam-4412	357	5	t1	t1	NOUN
ejpam-4412	357	6	is	be	AUX
ejpam-4412	357	7	normal	normal	ADJ
ejpam-4412	357	8	and	and	CCONJ
ejpam-4412	357	9	t2	t2	NOUN
ejpam-4412	357	10	is	be	AUX
ejpam-4412	357	11	pure	pure	ADJ
ejpam-4412	357	12	and	and	CCONJ
ejpam-4412	357	13	class	class	NOUN
ejpam-4412	357	14	p	p	NOUN
ejpam-4412	357	15	-	-	PUNCT
ejpam-4412	357	16	wa(s	wa(s	NUM
ejpam-4412	357	17	,	,	PUNCT
ejpam-4412	357	18	t	t	PROPN
ejpam-4412	357	19	)	)	PUNCT
ejpam-4412	357	20	,	,	PUNCT
ejpam-4412	357	21	i.e.	i.e.	X
ejpam-4412	357	22	,	,	PUNCT
ejpam-4412	357	23	t2	t2	PROPN
ejpam-4412	357	24	has	have	VERB
ejpam-4412	357	25	no	no	DET
ejpam-4412	357	26	invariant	invariant	ADJ
ejpam-4412	357	27	subspace	subspace	NOUN
ejpam-4412	357	28	m	m	VERB
ejpam-4412	357	29	such	such	ADJ
ejpam-4412	357	30	that	that	SCONJ
ejpam-4412	357	31	t2|m	t2|m	NOUN
ejpam-4412	357	32	is	be	AUX
ejpam-4412	357	33	normal	normal	ADJ
ejpam-4412	357	34	.	.	PUNCT
ejpam-4412	358	1	the	the	DET
ejpam-4412	358	2	next	next	ADJ
ejpam-4412	358	3	result	result	NOUN
ejpam-4412	358	4	was	be	AUX
ejpam-4412	358	5	proved	prove	VERB
ejpam-4412	358	6	for	for	ADP
ejpam-4412	358	7	dominant	dominant	ADJ
ejpam-4412	358	8	operators	operator	NOUN
ejpam-4412	358	9	in	in	ADP
ejpam-4412	358	10	[	[	X
ejpam-4412	358	11	28	28	NUM
ejpam-4412	358	12	,	,	PUNCT
ejpam-4412	358	13	theorem	theorem	VERB
ejpam-4412	358	14	1	1	NUM
ejpam-4412	358	15	]	]	PUNCT
ejpam-4412	358	16	,	,	PUNCT
ejpam-4412	358	17	for	for	ADP
ejpam-4412	358	18	p	p	PROPN
ejpam-4412	358	19	-	-	PUNCT
ejpam-4412	358	20	hyponormal	hyponormal	ADJ
ejpam-4412	358	21	operators	operator	NOUN
ejpam-4412	358	22	in	in	ADP
ejpam-4412	358	23	[	[	X
ejpam-4412	358	24	20	20	NUM
ejpam-4412	358	25	]	]	PUNCT
ejpam-4412	358	26	and	and	CCONJ
ejpam-4412	358	27	for	for	ADP
ejpam-4412	358	28	w	w	NOUN
ejpam-4412	358	29	-	-	PUNCT
ejpam-4412	358	30	hyponormal	hyponormal	ADJ
ejpam-4412	358	31	operators	operator	NOUN
ejpam-4412	358	32	in	in	ADP
ejpam-4412	358	33	[	[	X
ejpam-4412	358	34	22	22	NUM
ejpam-4412	358	35	,	,	PUNCT
ejpam-4412	358	36	lemma	lemma	PROPN
ejpam-4412	358	37	2.12	2.12	NUM
ejpam-4412	358	38	]	]	PUNCT
ejpam-4412	358	39	.	.	PUNCT
ejpam-4412	359	1	proposition	proposition	NOUN
ejpam-4412	359	2	2	2	NUM
ejpam-4412	359	3	.	.	PUNCT
ejpam-4412	360	1	let	let	AUX
ejpam-4412	360	2	t	t	PROPN
ejpam-4412	360	3	∈	∈	PROPN
ejpam-4412	360	4	b(h	b(h	PROPN
ejpam-4412	360	5	)	)	PUNCT
ejpam-4412	360	6	be	be	AUX
ejpam-4412	360	7	class	class	NOUN
ejpam-4412	360	8	p	p	NOUN
ejpam-4412	360	9	-	-	PUNCT
ejpam-4412	360	10	wa(s	wa(s	NUM
ejpam-4412	360	11	,	,	PUNCT
ejpam-4412	360	12	t)operator	t)operator	NOUN
ejpam-4412	360	13	for	for	ADP
ejpam-4412	360	14	0	0	NUM
ejpam-4412	360	15	<	<	X
ejpam-4412	360	16	s	s	PROPN
ejpam-4412	360	17	,	,	PUNCT
ejpam-4412	360	18	t	t	PROPN
ejpam-4412	360	19	,	,	PUNCT
ejpam-4412	360	20	s	s	PART
ejpam-4412	360	21	+	+	NUM
ejpam-4412	360	22	t	t	X
ejpam-4412	360	23	=	=	SYM
ejpam-4412	360	24	1	1	NUM
ejpam-4412	360	25	and	and	CCONJ
ejpam-4412	360	26	0	0	NUM
ejpam-4412	361	1	<	<	X
ejpam-4412	361	2	p	p	X
ejpam-4412	361	3	≤	≤	NUM
ejpam-4412	361	4	1	1	NUM
ejpam-4412	361	5	such	such	ADJ
ejpam-4412	361	6	that	that	DET
ejpam-4412	361	7	ker(t	ker(t	NOUN
ejpam-4412	361	8	)	)	PUNCT
ejpam-4412	361	9	⊂	⊂	PROPN
ejpam-4412	361	10	ker(t	ker(t	NOUN
ejpam-4412	361	11	∗	∗	NOUN
ejpam-4412	361	12	)	)	PUNCT
ejpam-4412	361	13	and	and	CCONJ
ejpam-4412	361	14	let	let	VERB
ejpam-4412	361	15	s	s	PRON
ejpam-4412	361	16	∈	∈	PROPN
ejpam-4412	361	17	b(k	b(k	PROPN
ejpam-4412	361	18	)	)	PUNCT
ejpam-4412	361	19	be	be	AUX
ejpam-4412	361	20	a	a	DET
ejpam-4412	361	21	normal	normal	ADJ
ejpam-4412	361	22	operator	operator	NOUN
ejpam-4412	361	23	.	.	PUNCT
ejpam-4412	362	1	if	if	SCONJ
ejpam-4412	362	2	there	there	PRON
ejpam-4412	362	3	exists	exist	VERB
ejpam-4412	362	4	a	a	DET
ejpam-4412	362	5	quasiaffinity	quasiaffinity	NOUN
ejpam-4412	362	6	x	x	SYM
ejpam-4412	362	7	∈	∈	PROPN
ejpam-4412	362	8	b(k	b(k	PROPN
ejpam-4412	362	9	,	,	PUNCT
ejpam-4412	362	10	h	h	NOUN
ejpam-4412	362	11	)	)	PUNCT
ejpam-4412	362	12	with	with	ADP
ejpam-4412	362	13	dense	dense	ADJ
ejpam-4412	362	14	range	range	NOUN
ejpam-4412	362	15	such	such	ADJ
ejpam-4412	362	16	that	that	SCONJ
ejpam-4412	362	17	tx	tx	PROPN
ejpam-4412	362	18	=	=	SYM
ejpam-4412	362	19	xs	xs	PROPN
ejpam-4412	362	20	,	,	PUNCT
ejpam-4412	362	21	then	then	ADV
ejpam-4412	362	22	t	t	PROPN
ejpam-4412	362	23	is	be	AUX
ejpam-4412	362	24	normal	normal	ADJ
ejpam-4412	362	25	.	.	PUNCT
ejpam-4412	363	1	to	to	PART
ejpam-4412	363	2	prove	prove	VERB
ejpam-4412	363	3	proposition	proposition	NOUN
ejpam-4412	363	4	2	2	NUM
ejpam-4412	363	5	,	,	PUNCT
ejpam-4412	363	6	we	we	PRON
ejpam-4412	363	7	need	need	VERB
ejpam-4412	363	8	the	the	DET
ejpam-4412	363	9	following	follow	VERB
ejpam-4412	363	10	lemmas	lemmas	PROPN
ejpam-4412	363	11	.	.	PUNCT
ejpam-4412	364	1	lemma	lemma	PROPN
ejpam-4412	364	2	11	11	NUM
ejpam-4412	364	3	.	.	PUNCT
ejpam-4412	365	1	[	[	X
ejpam-4412	365	2	9	9	NUM
ejpam-4412	365	3	]	]	X
ejpam-4412	365	4	if	if	SCONJ
ejpam-4412	365	5	n	n	PRON
ejpam-4412	365	6	is	be	AUX
ejpam-4412	365	7	a	a	DET
ejpam-4412	365	8	normal	normal	ADJ
ejpam-4412	365	9	operator	operator	NOUN
ejpam-4412	365	10	on	on	ADP
ejpam-4412	365	11	h	h	NOUN
ejpam-4412	365	12	,	,	PUNCT
ejpam-4412	365	13	then	then	ADV
ejpam-4412	365	14	we	we	PRON
ejpam-4412	365	15	have⋂	have⋂	ADV
ejpam-4412	365	16	λ∈c	λ∈c	NOUN
ejpam-4412	365	17	(	(	PUNCT
ejpam-4412	365	18	n	n	CCONJ
ejpam-4412	365	19	−	−	PROPN
ejpam-4412	365	20	λ)h	λ)h	X
ejpam-4412	365	21	=	=	PUNCT
ejpam-4412	365	22	{	{	PUNCT
ejpam-4412	365	23	0	0	NUM
ejpam-4412	365	24	}	}	PUNCT
ejpam-4412	365	25	.	.	PUNCT
ejpam-4412	366	1	lemma	lemma	PROPN
ejpam-4412	366	2	12	12	NUM
ejpam-4412	366	3	.	.	PUNCT
ejpam-4412	367	1	(	(	PUNCT
ejpam-4412	367	2	[	[	X
ejpam-4412	367	3	10	10	NUM
ejpam-4412	367	4	]	]	PUNCT
ejpam-4412	367	5	)	)	PUNCT
ejpam-4412	367	6	let	let	VERB
ejpam-4412	367	7	t	t	PROPN
ejpam-4412	367	8	∈	∈	PROPN
ejpam-4412	367	9	b(h	b(h	PROPN
ejpam-4412	367	10	)	)	PUNCT
ejpam-4412	367	11	,	,	PUNCT
ejpam-4412	368	1	d	d	PROPN
ejpam-4412	368	2	∈	∈	PROPN
ejpam-4412	368	3	b(h	b(h	PROPN
ejpam-4412	368	4	)	)	PUNCT
ejpam-4412	368	5	with	with	ADP
ejpam-4412	368	6	0	0	NUM
ejpam-4412	368	7	≤	≤	NUM
ejpam-4412	368	8	d	d	NOUN
ejpam-4412	368	9	≤	≤	NUM
ejpam-4412	368	10	m(t	m(t	NOUN
ejpam-4412	368	11	−	−	PROPN
ejpam-4412	368	12	λ)(t	λ)(t	PUNCT
ejpam-4412	368	13	−	−	PUNCT
ejpam-4412	369	1	λ)∗	λ)∗	ADP
ejpam-4412	369	2	for	for	ADP
ejpam-4412	369	3	all	all	DET
ejpam-4412	369	4	λ	λ	PROPN
ejpam-4412	369	5	∈	∈	PROPN
ejpam-4412	369	6	c	c	NOUN
ejpam-4412	369	7	,	,	PUNCT
ejpam-4412	369	8	where	where	SCONJ
ejpam-4412	369	9	m	m	NOUN
ejpam-4412	369	10	is	be	AUX
ejpam-4412	369	11	a	a	DET
ejpam-4412	369	12	positive	positive	ADJ
ejpam-4412	369	13	real	real	ADJ
ejpam-4412	369	14	number	number	NOUN
ejpam-4412	369	15	.	.	PUNCT
ejpam-4412	370	1	then	then	ADV
ejpam-4412	370	2	for	for	ADP
ejpam-4412	370	3	every	every	DET
ejpam-4412	370	4	x	x	SYM
ejpam-4412	370	5	∈	∈	PROPN
ejpam-4412	370	6	d	d	NOUN
ejpam-4412	370	7	1	1	NUM
ejpam-4412	370	8	2h	2h	NUM
ejpam-4412	370	9	there	there	ADV
ejpam-4412	370	10	exists	exist	VERB
ejpam-4412	370	11	a	a	DET
ejpam-4412	370	12	bounded	bounded	ADJ
ejpam-4412	370	13	function	function	NOUN
ejpam-4412	370	14	f	f	NOUN
ejpam-4412	370	15	:	:	PUNCT
ejpam-4412	370	16	c	c	VERB
ejpam-4412	371	1	−→	−→	ADJ
ejpam-4412	371	2	h	h	NOUN
ejpam-4412	371	3	such	such	ADJ
ejpam-4412	371	4	that	that	PRON
ejpam-4412	371	5	(	(	PUNCT
ejpam-4412	371	6	t	t	PROPN
ejpam-4412	371	7	−	−	PROPN
ejpam-4412	371	8	λ)f(λ	λ)f(λ	NOUN
ejpam-4412	371	9	)	)	PUNCT
ejpam-4412	371	10	≡	≡	PROPN
ejpam-4412	371	11	x.	x.	NOUN
ejpam-4412	371	12	proof	proof	NOUN
ejpam-4412	371	13	.	.	PUNCT
ejpam-4412	372	1	[	[	X
ejpam-4412	372	2	proof	proof	NOUN
ejpam-4412	372	3	of	of	ADP
ejpam-4412	372	4	proposition	proposition	NOUN
ejpam-4412	372	5	2	2	NUM
ejpam-4412	372	6	]	]	SYM
ejpam-4412	372	7	ker(t	ker(t	NOUN
ejpam-4412	372	8	)	)	PUNCT
ejpam-4412	372	9	⊂	⊂	PROPN
ejpam-4412	372	10	ker(t	ker(t	NOUN
ejpam-4412	372	11	∗	∗	NOUN
ejpam-4412	372	12	)	)	PUNCT
ejpam-4412	372	13	implies	imply	VERB
ejpam-4412	372	14	ker(t	ker(t	NOUN
ejpam-4412	372	15	)	)	PUNCT
ejpam-4412	372	16	reduces	reduce	VERB
ejpam-4412	372	17	t	t	NOUN
ejpam-4412	372	18	.	.	PUNCT
ejpam-4412	373	1	also	also	ADV
ejpam-4412	373	2	ker(s	ker(s	PROPN
ejpam-4412	373	3	)	)	PUNCT
ejpam-4412	373	4	reduces	reduce	VERB
ejpam-4412	373	5	s	s	PRON
ejpam-4412	373	6	since	since	SCONJ
ejpam-4412	373	7	s	s	PRON
ejpam-4412	373	8	is	be	AUX
ejpam-4412	373	9	normal	normal	ADJ
ejpam-4412	373	10	.	.	PUNCT
ejpam-4412	374	1	using	use	VERB
ejpam-4412	374	2	the	the	DET
ejpam-4412	374	3	orthogonal	orthogonal	ADJ
ejpam-4412	374	4	decompositions	decomposition	NOUN
ejpam-4412	374	5	h	h	NOUN
ejpam-4412	374	6	=	=	PUNCT
ejpam-4412	374	7	ran(|t	ran(|t	ADV
ejpam-4412	374	8	|)⊕	|)⊕	ADJ
ejpam-4412	374	9	ker(t	ker(t	NOUN
ejpam-4412	374	10	)	)	PUNCT
ejpam-4412	374	11	and	and	CCONJ
ejpam-4412	374	12	h	h	NOUN
ejpam-4412	374	13	=	=	SYM
ejpam-4412	374	14	ran(s	ran(s	PROPN
ejpam-4412	374	15	)	)	PUNCT
ejpam-4412	374	16	⊕	⊕	PROPN
ejpam-4412	374	17	ker(s	ker(s	PROPN
ejpam-4412	374	18	)	)	PUNCT
ejpam-4412	374	19	,	,	PUNCT
ejpam-4412	374	20	we	we	PRON
ejpam-4412	374	21	can	can	AUX
ejpam-4412	374	22	represent	represent	VERB
ejpam-4412	374	23	t	t	PROPN
ejpam-4412	374	24	and	and	CCONJ
ejpam-4412	374	25	s	s	PRON
ejpam-4412	374	26	as	as	SCONJ
ejpam-4412	374	27	follows	follow	VERB
ejpam-4412	374	28	:	:	PUNCT
ejpam-4412	374	29	t	t	NOUN
ejpam-4412	374	30	=	=	SYM
ejpam-4412	374	31	(	(	PUNCT
ejpam-4412	374	32	t1	t1	NOUN
ejpam-4412	374	33	0	0	NUM
ejpam-4412	374	34	0	0	NUM
ejpam-4412	374	35	0	0	NUM
ejpam-4412	374	36	)	)	PUNCT
ejpam-4412	374	37	,	,	PUNCT
ejpam-4412	374	38	s	s	X
ejpam-4412	374	39	=	=	PUNCT
ejpam-4412	374	40	(	(	PUNCT
ejpam-4412	374	41	s1	s1	NOUN
ejpam-4412	374	42	0	0	NUM
ejpam-4412	374	43	0	0	NUM
ejpam-4412	374	44	0	0	NUM
ejpam-4412	374	45	)	)	PUNCT
ejpam-4412	374	46	,	,	PUNCT
ejpam-4412	374	47	where	where	SCONJ
ejpam-4412	374	48	t1	t1	NOUN
ejpam-4412	374	49	is	be	AUX
ejpam-4412	374	50	an	an	DET
ejpam-4412	374	51	injective	injective	ADJ
ejpam-4412	374	52	class	class	NOUN
ejpam-4412	374	53	p	p	NOUN
ejpam-4412	374	54	-	-	PUNCT
ejpam-4412	374	55	wa(s	wa(s	NUM
ejpam-4412	374	56	,	,	PUNCT
ejpam-4412	374	57	t	t	NOUN
ejpam-4412	374	58	)	)	PUNCT
ejpam-4412	374	59	operator	operator	NOUN
ejpam-4412	374	60	on	on	ADP
ejpam-4412	374	61	ran(|t	ran(|t	ADV
ejpam-4412	374	62	|	|	ADV
ejpam-4412	374	63	)	)	PUNCT
ejpam-4412	374	64	and	and	CCONJ
ejpam-4412	374	65	s1	s1	NOUN
ejpam-4412	374	66	is	be	AUX
ejpam-4412	374	67	injective	injective	ADJ
ejpam-4412	374	68	normal	normal	ADJ
ejpam-4412	374	69	on	on	ADP
ejpam-4412	374	70	ran(s	ran(s	PROPN
ejpam-4412	374	71	)	)	PUNCT
ejpam-4412	374	72	.	.	PUNCT
ejpam-4412	375	1	the	the	DET
ejpam-4412	375	2	assumption	assumption	NOUN
ejpam-4412	375	3	tx	tx	PROPN
ejpam-4412	375	4	=	=	SYM
ejpam-4412	375	5	xs	xs	PROPN
ejpam-4412	375	6	asserts	assert	VERB
ejpam-4412	375	7	that	that	SCONJ
ejpam-4412	375	8	x	x	PROPN
ejpam-4412	375	9	maps	maps	PROPN
ejpam-4412	375	10	ran(s	ran(s	PROPN
ejpam-4412	375	11	)	)	PUNCT
ejpam-4412	375	12	to	to	ADP
ejpam-4412	375	13	ran(t	ran(t	PROPN
ejpam-4412	375	14	)	)	PUNCT
ejpam-4412	376	1	⊂	⊂	PROPN
ejpam-4412	377	1	ran(|t	ran(|t	ADV
ejpam-4412	377	2	|	|	ADV
ejpam-4412	377	3	)	)	PUNCT
ejpam-4412	377	4	and	and	CCONJ
ejpam-4412	377	5	ker(s	ker(s	NOUN
ejpam-4412	377	6	)	)	PUNCT
ejpam-4412	377	7	to	to	ADP
ejpam-4412	377	8	ker(t	ker(t	PROPN
ejpam-4412	377	9	)	)	PUNCT
ejpam-4412	377	10	,	,	PUNCT
ejpam-4412	377	11	hence	hence	ADV
ejpam-4412	377	12	x	x	PUNCT
ejpam-4412	377	13	is	be	AUX
ejpam-4412	377	14	the	the	DET
ejpam-4412	377	15	form	form	NOUN
ejpam-4412	377	16	:	:	PUNCT
ejpam-4412	377	17	x	x	SYM
ejpam-4412	377	18	=	=	PUNCT
ejpam-4412	377	19	(	(	PUNCT
ejpam-4412	377	20	x1	x1	PROPN
ejpam-4412	377	21	0	0	NUM
ejpam-4412	377	22	0	0	NUM
ejpam-4412	377	23	x2	x2	PROPN
ejpam-4412	377	24	)	)	PUNCT
ejpam-4412	377	25	,	,	PUNCT
ejpam-4412	377	26	where	where	SCONJ
ejpam-4412	377	27	x1	x1	PROPN
ejpam-4412	377	28	∈	∈	PROPN
ejpam-4412	377	29	b(ran(s	b(ran(s	PROPN
ejpam-4412	377	30	)	)	PUNCT
ejpam-4412	377	31	,	,	PUNCT
ejpam-4412	377	32	ran(|t	ran(|t	ADV
ejpam-4412	377	33	|	|	ADV
ejpam-4412	377	34	)	)	PUNCT
ejpam-4412	377	35	,	,	PUNCT
ejpam-4412	377	36	x2	x2	PROPN
ejpam-4412	377	37	∈	∈	PROPN
ejpam-4412	377	38	b(ker(s	b(ker(s	PROPN
ejpam-4412	377	39	)	)	PUNCT
ejpam-4412	377	40	,	,	PUNCT
ejpam-4412	377	41	ker(t	ker(t	NOUN
ejpam-4412	377	42	)	)	PUNCT
ejpam-4412	377	43	)	)	PUNCT
ejpam-4412	377	44	.	.	PUNCT
ejpam-4412	378	1	since	since	SCONJ
ejpam-4412	378	2	tx	tx	PROPN
ejpam-4412	378	3	=	=	SYM
ejpam-4412	378	4	xs	xs	PROPN
ejpam-4412	378	5	,	,	PUNCT
ejpam-4412	378	6	we	we	PRON
ejpam-4412	378	7	have	have	VERB
ejpam-4412	378	8	that	that	PRON
ejpam-4412	378	9	t1x1	t1x1	NOUN
ejpam-4412	378	10	=	=	SYM
ejpam-4412	378	11	x1s1	x1s1	PROPN
ejpam-4412	378	12	.	.	PUNCT
ejpam-4412	379	1	since	since	SCONJ
ejpam-4412	379	2	x	x	PRON
ejpam-4412	379	3	is	be	AUX
ejpam-4412	379	4	injective	injective	ADJ
ejpam-4412	379	5	with	with	ADP
ejpam-4412	379	6	dense	dense	ADJ
ejpam-4412	379	7	range	range	NOUN
ejpam-4412	379	8	,	,	PUNCT
ejpam-4412	379	9	x1	x1	PROPN
ejpam-4412	379	10	is	be	AUX
ejpam-4412	379	11	also	also	ADV
ejpam-4412	379	12	injective	injective	ADJ
ejpam-4412	379	13	with	with	ADP
ejpam-4412	379	14	dense	dense	ADJ
ejpam-4412	379	15	range	range	NOUN
ejpam-4412	379	16	.	.	PUNCT
ejpam-4412	380	1	put	put	VERB
ejpam-4412	380	2	w1	w1	NOUN
ejpam-4412	380	3	=	=	SYM
ejpam-4412	380	4	|t1|sx1	|t1|sx1	PROPN
ejpam-4412	380	5	,	,	PUNCT
ejpam-4412	380	6	then	then	ADV
ejpam-4412	380	7	w1	w1	NOUN
ejpam-4412	380	8	is	be	AUX
ejpam-4412	380	9	also	also	ADV
ejpam-4412	380	10	injective	injective	ADJ
ejpam-4412	380	11	with	with	ADP
ejpam-4412	380	12	dense	dense	ADJ
ejpam-4412	380	13	range	range	NOUN
ejpam-4412	380	14	and	and	CCONJ
ejpam-4412	380	15	satisfies	satisfie	NOUN
ejpam-4412	380	16	t	t	PROPN
ejpam-4412	380	17	(	(	PUNCT
ejpam-4412	380	18	s	s	PROPN
ejpam-4412	380	19	,	,	PUNCT
ejpam-4412	380	20	t)w1	t)w1	PROPN
ejpam-4412	380	21	=	=	SYM
ejpam-4412	380	22	w1s	w1s	PROPN
ejpam-4412	380	23	.	.	PUNCT
ejpam-4412	381	1	put	put	VERB
ejpam-4412	381	2	wn	wn	PROPN
ejpam-4412	382	1	=	=	PUNCT
ejpam-4412	382	2	|∆n(t	|∆n(t	PROPN
ejpam-4412	382	3	(	(	PUNCT
ejpam-4412	382	4	s	s	PROPN
ejpam-4412	382	5	,	,	PUNCT
ejpam-4412	382	6	t))|swn−1	t))|swn−1	PROPN
ejpam-4412	382	7	,	,	PUNCT
ejpam-4412	382	8	then	then	ADV
ejpam-4412	382	9	wn	wn	PROPN
ejpam-4412	382	10	is	be	AUX
ejpam-4412	382	11	also	also	ADV
ejpam-4412	382	12	injective	injective	ADJ
ejpam-4412	382	13	with	with	ADP
ejpam-4412	382	14	dense	dense	ADJ
ejpam-4412	382	15	range	range	NOUN
ejpam-4412	382	16	and	and	CCONJ
ejpam-4412	382	17	satisfies	satisfie	NOUN
ejpam-4412	382	18	∆n(t	∆n(t	NUM
ejpam-4412	382	19	(	(	PUNCT
ejpam-4412	382	20	s	s	PROPN
ejpam-4412	382	21	,	,	PUNCT
ejpam-4412	382	22	t))wn	t))wn	NOUN
ejpam-4412	382	23	=	=	SYM
ejpam-4412	382	24	wns	wns	PROPN
ejpam-4412	382	25	.	.	PUNCT
ejpam-4412	383	1	from	from	ADP
ejpam-4412	383	2	[	[	X
ejpam-4412	383	3	26	26	NUM
ejpam-4412	383	4	,	,	PUNCT
ejpam-4412	383	5	corollary	corollary	ADJ
ejpam-4412	383	6	2.7	2.7	NUM
ejpam-4412	383	7	]	]	PUNCT
ejpam-4412	383	8	and	and	CCONJ
ejpam-4412	383	9	[	[	X
ejpam-4412	383	10	6	6	NUM
ejpam-4412	383	11	]	]	PUNCT
ejpam-4412	383	12	,	,	PUNCT
ejpam-4412	383	13	if	if	SCONJ
ejpam-4412	383	14	there	there	PRON
ejpam-4412	383	15	exists	exist	VERB
ejpam-4412	383	16	an	an	DET
ejpam-4412	383	17	integer	integer	NOUN
ejpam-4412	383	18	m	m	VERB
ejpam-4412	383	19	such	such	ADJ
ejpam-4412	383	20	that	that	SCONJ
ejpam-4412	383	21	∆m(t	∆m(t	PROPN
ejpam-4412	383	22	(	(	PUNCT
ejpam-4412	383	23	s	s	PROPN
ejpam-4412	383	24	,	,	PUNCT
ejpam-4412	383	25	t	t	PROPN
ejpam-4412	383	26	)	)	PUNCT
ejpam-4412	383	27	)	)	PUNCT
ejpam-4412	383	28	is	be	AUX
ejpam-4412	383	29	a	a	DET
ejpam-4412	383	30	hyponormal	hyponormal	ADJ
ejpam-4412	383	31	operator	operator	NOUN
ejpam-4412	383	32	,	,	PUNCT
ejpam-4412	383	33	then	then	ADV
ejpam-4412	383	34	∆n(t	∆n(t	PROPN
ejpam-4412	383	35	(	(	PUNCT
ejpam-4412	383	36	s	s	PROPN
ejpam-4412	383	37	,	,	PUNCT
ejpam-4412	383	38	t	t	PROPN
ejpam-4412	383	39	)	)	PUNCT
ejpam-4412	383	40	)	)	PUNCT
ejpam-4412	383	41	is	be	AUX
ejpam-4412	383	42	a	a	DET
ejpam-4412	383	43	hyponormal	hyponormal	ADJ
ejpam-4412	383	44	operator	operator	NOUN
ejpam-4412	383	45	for	for	ADP
ejpam-4412	383	46	n	n	PRON
ejpam-4412	383	47	≥	≥	NOUN
ejpam-4412	383	48	m.	m.	NOUN
ejpam-4412	383	49	it	it	PRON
ejpam-4412	383	50	follows	follow	VERB
ejpam-4412	383	51	from	from	ADP
ejpam-4412	383	52	lemma	lemma	PROPN
ejpam-4412	383	53	12	12	NUM
ejpam-4412	383	54	that	that	SCONJ
ejpam-4412	383	55	there	there	PRON
ejpam-4412	383	56	exists	exist	VERB
ejpam-4412	383	57	a	a	DET
ejpam-4412	383	58	bounded	bounded	ADJ
ejpam-4412	383	59	function	function	NOUN
ejpam-4412	383	60	f	f	NOUN
ejpam-4412	383	61	:	:	PUNCT
ejpam-4412	384	1	c	c	VERB
ejpam-4412	384	2	−→	−→	ADJ
ejpam-4412	384	3	h	h	NOUN
ejpam-4412	384	4	such	such	ADJ
ejpam-4412	384	5	that	that	PRON
ejpam-4412	384	6	(	(	PUNCT
ejpam-4412	384	7	∆n(t1(s	∆n(t1(s	PROPN
ejpam-4412	384	8	,	,	PUNCT
ejpam-4412	384	9	t	t	PROPN
ejpam-4412	384	10	)	)	PUNCT
ejpam-4412	384	11	)	)	PUNCT
ejpam-4412	384	12	∗	∗	NOUN
ejpam-4412	384	13	−	−	PROPN
ejpam-4412	384	14	λ)f(λ	λ)f(λ	NOUN
ejpam-4412	384	15	)	)	PUNCT
ejpam-4412	384	16	≡	≡	PROPN
ejpam-4412	384	17	x	x	PROPN
ejpam-4412	384	18	,	,	PUNCT
ejpam-4412	384	19	for	for	ADP
ejpam-4412	384	20	every	every	DET
ejpam-4412	384	21	x	x	SYM
ejpam-4412	384	22	∈	∈	PROPN
ejpam-4412	384	23	(	(	PUNCT
ejpam-4412	384	24	∆n(t1(s	∆n(t1(s	PROPN
ejpam-4412	384	25	,	,	PUNCT
ejpam-4412	384	26	t	t	PROPN
ejpam-4412	384	27	)	)	PUNCT
ejpam-4412	384	28	)	)	PUNCT
ejpam-4412	384	29	∗∆n(t1(s	∗∆n(t1(s	PROPN
ejpam-4412	384	30	,	,	PUNCT
ejpam-4412	384	31	t)−	t)−	PROPN
ejpam-4412	384	32	∆n(t1(s	∆n(t1(	VERB
ejpam-4412	384	33	,	,	PUNCT
ejpam-4412	384	34	t)(∆	t)(∆	NUM
ejpam-4412	384	35	n(t1(s	n(t1(s	PROPN
ejpam-4412	384	36	,	,	PUNCT
ejpam-4412	384	37	t	t	PROPN
ejpam-4412	384	38	)	)	PUNCT
ejpam-4412	384	39	)	)	PUNCT
ejpam-4412	385	1	∗	∗	NOUN
ejpam-4412	385	2	)	)	PUNCT
ejpam-4412	385	3	1	1	NUM
ejpam-4412	385	4	2h	2h	NUM
ejpam-4412	385	5	.	.	PUNCT
ejpam-4412	386	1	hence	hence	ADV
ejpam-4412	386	2	w	w	ADP
ejpam-4412	386	3	∗	∗	NOUN
ejpam-4412	386	4	nx	nx	NOUN
ejpam-4412	387	1	=	=	SYM
ejpam-4412	387	2	w	w	PROPN
ejpam-4412	387	3	∗	∗	NOUN
ejpam-4412	387	4	n(∆	n(∆	PROPN
ejpam-4412	387	5	n(t1(s	n(t1(s	PROPN
ejpam-4412	387	6	,	,	PUNCT
ejpam-4412	387	7	t	t	PROPN
ejpam-4412	387	8	)	)	PUNCT
ejpam-4412	387	9	)	)	PUNCT
ejpam-4412	387	10	∗	∗	NOUN
ejpam-4412	387	11	−	−	PROPN
ejpam-4412	387	12	λ)f(λ	λ)f(λ	NOUN
ejpam-4412	387	13	)	)	PUNCT
ejpam-4412	387	14	=	=	PUNCT
ejpam-4412	387	15	(	(	PUNCT
ejpam-4412	387	16	s∗	s∗	PROPN
ejpam-4412	387	17	1	1	NUM
ejpam-4412	387	18	−	−	NOUN
ejpam-4412	387	19	λ)w	λ)w	PUNCT
ejpam-4412	387	20	∗	∗	NOUN
ejpam-4412	387	21	nf(λ	nf(λ	X
ejpam-4412	387	22	)	)	PUNCT
ejpam-4412	387	23	∈	∈	NOUN
ejpam-4412	387	24	ran(s∗	ran(s∗	NOUN
ejpam-4412	387	25	1	1	NUM
ejpam-4412	387	26	−	−	PROPN
ejpam-4412	387	27	λ	λ	NOUN
ejpam-4412	387	28	)	)	PUNCT
ejpam-4412	387	29	for	for	ADP
ejpam-4412	387	30	all	all	DET
ejpam-4412	387	31	λ	λ	PROPN
ejpam-4412	387	32	∈	∈	PROPN
ejpam-4412	387	33	c.	c.	PROPN
ejpam-4412	387	34	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	387	35	,	,	PUNCT
ejpam-4412	387	36	n.	n.	NOUN
ejpam-4412	387	37	h.	h.	PROPN
ejpam-4412	387	38	altaweel	altaweel	PROPN
ejpam-4412	387	39	/	/	SYM
ejpam-4412	387	40	eur	eur	PROPN
ejpam-4412	387	41	.	.	PUNCT
ejpam-4412	388	1	j.	j.	PROPN
ejpam-4412	388	2	pure	pure	PROPN
ejpam-4412	388	3	appl	appl	PROPN
ejpam-4412	388	4	.	.	PROPN
ejpam-4412	388	5	math	math	PROPN
ejpam-4412	388	6	,	,	PUNCT
ejpam-4412	388	7	15	15	NUM
ejpam-4412	388	8	(	(	PUNCT
ejpam-4412	388	9	3	3	NUM
ejpam-4412	388	10	)	)	PUNCT
ejpam-4412	388	11	(	(	PUNCT
ejpam-4412	388	12	2022	2022	NUM
ejpam-4412	388	13	)	)	PUNCT
ejpam-4412	388	14	,	,	PUNCT
ejpam-4412	388	15	1067	1067	NUM
ejpam-4412	388	16	-	-	SYM
ejpam-4412	388	17	1089	1089	NUM
ejpam-4412	388	18	1080	1080	NUM
ejpam-4412	388	19	by	by	ADP
ejpam-4412	388	20	lemma	lemma	PROPN
ejpam-4412	388	21	11	11	NUM
ejpam-4412	389	1	,	,	PUNCT
ejpam-4412	389	2	we	we	PRON
ejpam-4412	389	3	have	have	VERB
ejpam-4412	389	4	w	w	PROPN
ejpam-4412	389	5	∗	∗	NOUN
ejpam-4412	389	6	nx	nx	X
ejpam-4412	390	1	=	=	SYM
ejpam-4412	390	2	0	0	NUM
ejpam-4412	390	3	,	,	PUNCT
ejpam-4412	390	4	and	and	CCONJ
ejpam-4412	391	1	hence	hence	ADV
ejpam-4412	391	2	x	x	PUNCT
ejpam-4412	391	3	=	=	NOUN
ejpam-4412	391	4	0	0	PUNCT
ejpam-4412	391	5	because	because	SCONJ
ejpam-4412	391	6	w	w	NOUN
ejpam-4412	391	7	∗	∗	NOUN
ejpam-4412	391	8	n	n	PART
ejpam-4412	391	9	is	be	AUX
ejpam-4412	391	10	injective	injective	ADJ
ejpam-4412	391	11	.	.	PUNCT
ejpam-4412	392	1	this	this	PRON
ejpam-4412	392	2	implies	imply	VERB
ejpam-4412	392	3	that	that	SCONJ
ejpam-4412	392	4	∆n(t1(s	∆n(t1(s	PROPN
ejpam-4412	392	5	,	,	PUNCT
ejpam-4412	392	6	t	t	PROPN
ejpam-4412	392	7	)	)	PUNCT
ejpam-4412	392	8	is	be	AUX
ejpam-4412	392	9	normal	normal	ADJ
ejpam-4412	392	10	.	.	PUNCT
ejpam-4412	393	1	by	by	ADP
ejpam-4412	393	2	corollary	corollary	ADJ
ejpam-4412	393	3	1	1	NUM
ejpam-4412	393	4	,	,	PUNCT
ejpam-4412	393	5	t1	t1	PROPN
ejpam-4412	393	6	is	be	AUX
ejpam-4412	393	7	normal	normal	ADJ
ejpam-4412	393	8	and	and	CCONJ
ejpam-4412	393	9	therefore	therefore	ADV
ejpam-4412	393	10	t	t	PROPN
ejpam-4412	393	11	=	=	SYM
ejpam-4412	393	12	t1	t1	PROPN
ejpam-4412	393	13	⊕	⊕	PROPN
ejpam-4412	393	14	0	0	NUM
ejpam-4412	393	15	is	be	AUX
ejpam-4412	393	16	also	also	ADV
ejpam-4412	393	17	normal	normal	ADJ
ejpam-4412	393	18	.	.	PUNCT
ejpam-4412	394	1	theorem	theorem	ADJ
ejpam-4412	394	2	8	8	NUM
ejpam-4412	394	3	.	.	PUNCT
ejpam-4412	395	1	let	let	VERB
ejpam-4412	395	2	t	t	PROPN
ejpam-4412	395	3	and	and	CCONJ
ejpam-4412	395	4	s∗	s∗	PROPN
ejpam-4412	395	5	be	be	AUX
ejpam-4412	395	6	class	class	NOUN
ejpam-4412	395	7	p	p	NOUN
ejpam-4412	395	8	-	-	PUNCT
ejpam-4412	395	9	wa(s	wa(s	NUM
ejpam-4412	395	10	,	,	PUNCT
ejpam-4412	395	11	t	t	NOUN
ejpam-4412	395	12	)	)	PUNCT
ejpam-4412	395	13	operators	operator	NOUN
ejpam-4412	395	14	with	with	ADP
ejpam-4412	395	15	0	0	NUM
ejpam-4412	395	16	<	<	X
ejpam-4412	395	17	s	s	PROPN
ejpam-4412	395	18	,	,	PUNCT
ejpam-4412	395	19	t	t	PROPN
ejpam-4412	395	20	,	,	PUNCT
ejpam-4412	395	21	s	s	PART
ejpam-4412	395	22	+	+	NUM
ejpam-4412	395	23	t	t	X
ejpam-4412	395	24	=	=	SYM
ejpam-4412	395	25	1	1	NUM
ejpam-4412	395	26	and	and	CCONJ
ejpam-4412	395	27	0	0	NUM
ejpam-4412	395	28	<	<	X
ejpam-4412	395	29	p	p	X
ejpam-4412	395	30	≤	≤	NUM
ejpam-4412	395	31	1	1	NUM
ejpam-4412	395	32	such	such	ADJ
ejpam-4412	395	33	that	that	DET
ejpam-4412	395	34	ker(t	ker(t	NOUN
ejpam-4412	395	35	)	)	PUNCT
ejpam-4412	396	1	⊂	⊂	PROPN
ejpam-4412	396	2	ker(t	ker(t	NOUN
ejpam-4412	396	3	∗	∗	NOUN
ejpam-4412	396	4	)	)	PUNCT
ejpam-4412	396	5	and	and	CCONJ
ejpam-4412	396	6	ker(s∗	ker(s∗	X
ejpam-4412	396	7	)	)	PUNCT
ejpam-4412	396	8	⊂	⊂	PROPN
ejpam-4412	397	1	ker(s	ker(s	PROPN
ejpam-4412	397	2	)	)	PUNCT
ejpam-4412	397	3	.	.	PUNCT
ejpam-4412	398	1	if	if	SCONJ
ejpam-4412	398	2	there	there	PRON
ejpam-4412	398	3	exist	exist	VERB
ejpam-4412	398	4	a	a	DET
ejpam-4412	398	5	quasiaffinity	quasiaffinity	NOUN
ejpam-4412	398	6	x	x	PUNCT
ejpam-4412	398	7	such	such	ADJ
ejpam-4412	398	8	that	that	DET
ejpam-4412	398	9	tx	tx	PROPN
ejpam-4412	398	10	=	=	SYM
ejpam-4412	398	11	xs	xs	PROPN
ejpam-4412	398	12	,	,	PUNCT
ejpam-4412	398	13	then	then	ADV
ejpam-4412	398	14	t	t	PROPN
ejpam-4412	398	15	and	and	CCONJ
ejpam-4412	398	16	s	s	PRON
ejpam-4412	398	17	are	be	AUX
ejpam-4412	398	18	unitarily	unitarily	ADV
ejpam-4412	398	19	equivalent	equivalent	ADJ
ejpam-4412	398	20	normal	normal	ADJ
ejpam-4412	398	21	operators	operator	NOUN
ejpam-4412	398	22	.	.	PUNCT
ejpam-4412	399	1	proof	proof	NOUN
ejpam-4412	399	2	.	.	PUNCT
ejpam-4412	400	1	first	first	ADJ
ejpam-4412	400	2	decompose	decompose	VERB
ejpam-4412	400	3	t	t	PROPN
ejpam-4412	400	4	and	and	CCONJ
ejpam-4412	400	5	s∗	s∗	VERB
ejpam-4412	400	6	into	into	ADP
ejpam-4412	400	7	their	their	PRON
ejpam-4412	400	8	normal	normal	ADJ
ejpam-4412	400	9	and	and	CCONJ
ejpam-4412	400	10	pure	pure	ADJ
ejpam-4412	400	11	parts	part	NOUN
ejpam-4412	400	12	by	by	ADP
ejpam-4412	400	13	t	t	PROPN
ejpam-4412	400	14	=	=	SYM
ejpam-4412	400	15	t1	t1	PROPN
ejpam-4412	400	16	⊕	⊕	PROPN
ejpam-4412	400	17	t2	t2	PROPN
ejpam-4412	400	18	on	on	ADP
ejpam-4412	400	19	h	h	NOUN
ejpam-4412	400	20	=	=	PUNCT
ejpam-4412	400	21	h1	h1	PROPN
ejpam-4412	400	22	⊕	⊕	PROPN
ejpam-4412	400	23	h2	h2	PROPN
ejpam-4412	400	24	and	and	CCONJ
ejpam-4412	400	25	s∗	s∗	PROPN
ejpam-4412	400	26	=	=	SYM
ejpam-4412	400	27	s∗	s∗	PROPN
ejpam-4412	400	28	1	1	NUM
ejpam-4412	400	29	⊕	⊕	PROPN
ejpam-4412	400	30	s∗	s∗	VERB
ejpam-4412	400	31	2	2	NUM
ejpam-4412	400	32	on	on	ADP
ejpam-4412	400	33	k	k	PROPN
ejpam-4412	400	34	=	=	SYM
ejpam-4412	400	35	k1	k1	PROPN
ejpam-4412	400	36	⊕	⊕	PROPN
ejpam-4412	400	37	k2	k2	PROPN
ejpam-4412	400	38	,	,	PUNCT
ejpam-4412	400	39	where	where	SCONJ
ejpam-4412	400	40	t1	t1	NOUN
ejpam-4412	400	41	,	,	PUNCT
ejpam-4412	400	42	s1	s1	PROPN
ejpam-4412	400	43	are	be	AUX
ejpam-4412	400	44	normal	normal	ADJ
ejpam-4412	400	45	and	and	CCONJ
ejpam-4412	400	46	t2	t2	NOUN
ejpam-4412	400	47	,	,	PUNCT
ejpam-4412	400	48	s	s	PART
ejpam-4412	400	49	∗	∗	NOUN
ejpam-4412	400	50	2	2	NUM
ejpam-4412	400	51	are	be	AUX
ejpam-4412	400	52	pure	pure	ADJ
ejpam-4412	400	53	.	.	PUNCT
ejpam-4412	401	1	let	let	VERB
ejpam-4412	401	2	x	x	PUNCT
ejpam-4412	401	3	=	=	PUNCT
ejpam-4412	402	1	[	[	X
ejpam-4412	402	2	xij	xij	X
ejpam-4412	402	3	]	]	PUNCT
ejpam-4412	402	4	2	2	NUM
ejpam-4412	402	5	i	i	NOUN
ejpam-4412	402	6	,	,	PUNCT
ejpam-4412	402	7	j=1	j=1	PROPN
ejpam-4412	402	8	.	.	PUNCT
ejpam-4412	403	1	then	then	ADV
ejpam-4412	403	2	tx	tx	PROPN
ejpam-4412	403	3	=	=	SYM
ejpam-4412	403	4	xs	xs	PROPN
ejpam-4412	403	5	implies	imply	VERB
ejpam-4412	403	6	that	that	SCONJ
ejpam-4412	403	7	t2x21	t2x21	PROPN
ejpam-4412	403	8	=	=	PUNCT
ejpam-4412	403	9	x21s1	x21s1	PROPN
ejpam-4412	403	10	and	and	CCONJ
ejpam-4412	403	11	t2x22	t2x22	PROPN
ejpam-4412	403	12	=	=	SYM
ejpam-4412	403	13	x22s2	x22s2	PROPN
ejpam-4412	403	14	.	.	PUNCT
ejpam-4412	404	1	let	let	VERB
ejpam-4412	404	2	t2	t2	PROPN
ejpam-4412	404	3	=	=	SYM
ejpam-4412	404	4	u2|t2|	u2|t2|	PROPN
ejpam-4412	404	5	,	,	PUNCT
ejpam-4412	404	6	s∗	s∗	PROPN
ejpam-4412	404	7	2	2	NUM
ejpam-4412	404	8	=	=	SYM
ejpam-4412	404	9	v	v	NOUN
ejpam-4412	404	10	∗	∗	NOUN
ejpam-4412	404	11	2	2	NUM
ejpam-4412	404	12	|s∗	|s∗	PROPN
ejpam-4412	404	13	2	2	NUM
ejpam-4412	404	14	|	|	ADV
ejpam-4412	404	15	be	be	AUX
ejpam-4412	404	16	the	the	DET
ejpam-4412	404	17	polar	polar	ADJ
ejpam-4412	404	18	decompositions	decomposition	NOUN
ejpam-4412	404	19	of	of	ADP
ejpam-4412	404	20	t2	t2	NOUN
ejpam-4412	404	21	and	and	CCONJ
ejpam-4412	404	22	s∗	s∗	PROPN
ejpam-4412	404	23	2	2	NUM
ejpam-4412	404	24	,	,	PUNCT
ejpam-4412	404	25	respectively	respectively	ADV
ejpam-4412	404	26	and	and	CCONJ
ejpam-4412	404	27	t2(s	t2(s	PROPN
ejpam-4412	404	28	,	,	PUNCT
ejpam-4412	404	29	t	t	PROPN
ejpam-4412	404	30	)	)	PUNCT
ejpam-4412	404	31	=	=	SYM
ejpam-4412	405	1	|t2|su2|t2|t	|t2|su2|t2|t	PROPN
ejpam-4412	405	2	,	,	PUNCT
ejpam-4412	405	3	s∗	s∗	VERB
ejpam-4412	405	4	2(s	2(s	NUM
ejpam-4412	405	5	,	,	PUNCT
ejpam-4412	405	6	t	t	PROPN
ejpam-4412	405	7	)	)	PUNCT
ejpam-4412	405	8	=	=	PUNCT
ejpam-4412	406	1	|s∗	|s∗	PROPN
ejpam-4412	406	2	2	2	NUM
ejpam-4412	406	3	|sv	|sv	NUM
ejpam-4412	406	4	∗	∗	NOUN
ejpam-4412	406	5	2	2	NUM
ejpam-4412	406	6	|s∗	|s∗	PROPN
ejpam-4412	406	7	2	2	NUM
ejpam-4412	406	8	|t	|t	PROPN
ejpam-4412	406	9	,	,	PUNCT
ejpam-4412	406	10	w	w	PROPN
ejpam-4412	406	11	=	=	PUNCT
ejpam-4412	406	12	|t2|sx22|s∗	|t2|sx22|s∗	NUM
ejpam-4412	406	13	2	2	NUM
ejpam-4412	406	14	|s	|s	PROPN
ejpam-4412	406	15	.	.	PUNCT
ejpam-4412	407	1	then	then	ADV
ejpam-4412	407	2	t2(s	t2(s	NOUN
ejpam-4412	407	3	,	,	PUNCT
ejpam-4412	407	4	t)w	t)w	X
ejpam-4412	407	5	=	=	PUNCT
ejpam-4412	407	6	|t2|st2x22|s∗	|t2|st2x22|s∗	PROPN
ejpam-4412	407	7	2	2	NUM
ejpam-4412	407	8	|s	|s	PROPN
ejpam-4412	407	9	=	=	NOUN
ejpam-4412	407	10	|t2|sx22s2|s∗	|t2|sx22s2|s∗	NOUN
ejpam-4412	407	11	2	2	NUM
ejpam-4412	407	12	|s	|s	PROPN
ejpam-4412	407	13	=	=	SYM
ejpam-4412	407	14	w	w	PROPN
ejpam-4412	407	15	(	(	PUNCT
ejpam-4412	407	16	s∗	s∗	PROPN
ejpam-4412	407	17	2(s	2(s	NUM
ejpam-4412	407	18	,	,	PUNCT
ejpam-4412	407	19	t	t	PROPN
ejpam-4412	407	20	)	)	PUNCT
ejpam-4412	407	21	)	)	PUNCT
ejpam-4412	408	1	∗.	∗.	PROPN
ejpam-4412	408	2	since	since	SCONJ
ejpam-4412	408	3	ran(w	ran(w	PROPN
ejpam-4412	408	4	)	)	PUNCT
ejpam-4412	408	5	reduces	reduce	VERB
ejpam-4412	408	6	t2(s	t2(s	PROPN
ejpam-4412	408	7	,	,	PUNCT
ejpam-4412	408	8	t	t	PROPN
ejpam-4412	408	9	)	)	PUNCT
ejpam-4412	408	10	and	and	CCONJ
ejpam-4412	408	11	ker(w	ker(w	PROPN
ejpam-4412	408	12	)	)	PUNCT
ejpam-4412	409	1	⊥	⊥	NOUN
ejpam-4412	409	2	reduces	reduce	VERB
ejpam-4412	409	3	s∗	s∗	PROPN
ejpam-4412	409	4	2(s	2(s	NUM
ejpam-4412	409	5	,	,	PUNCT
ejpam-4412	409	6	t	t	PROPN
ejpam-4412	409	7	)	)	PUNCT
ejpam-4412	409	8	and	and	CCONJ
ejpam-4412	409	9	t2(s	t2(s	NOUN
ejpam-4412	409	10	,	,	PUNCT
ejpam-4412	409	11	t)|ran(w	t)|ran(w	NUM
ejpam-4412	409	12	)	)	PUNCT
ejpam-4412	409	13	and	and	CCONJ
ejpam-4412	409	14	s∗	s∗	PROPN
ejpam-4412	409	15	2(s	2(s	NUM
ejpam-4412	409	16	,	,	PUNCT
ejpam-4412	409	17	t)|ker(w	t)|ker(w	PRON
ejpam-4412	409	18	)	)	PUNCT
ejpam-4412	410	1	⊥	⊥	NOUN
ejpam-4412	410	2	are	be	AUX
ejpam-4412	410	3	unitarily	unitarily	ADV
ejpam-4412	410	4	equivalent	equivalent	ADJ
ejpam-4412	410	5	normal	normal	ADJ
ejpam-4412	410	6	operators	operator	NOUN
ejpam-4412	410	7	,	,	PUNCT
ejpam-4412	410	8	and	and	CCONJ
ejpam-4412	410	9	since	since	SCONJ
ejpam-4412	410	10	t2	t2	NOUN
ejpam-4412	410	11	,	,	PUNCT
ejpam-4412	410	12	s	s	PART
ejpam-4412	410	13	∗	∗	NOUN
ejpam-4412	410	14	2	2	NUM
ejpam-4412	410	15	are	be	AUX
ejpam-4412	410	16	injective	injective	ADJ
ejpam-4412	410	17	class	class	NOUN
ejpam-4412	410	18	p	p	NOUN
ejpam-4412	410	19	-	-	PUNCT
ejpam-4412	410	20	wa(s	wa(s	NUM
ejpam-4412	410	21	,	,	PUNCT
ejpam-4412	410	22	t	t	NOUN
ejpam-4412	410	23	)	)	PUNCT
ejpam-4412	410	24	operators	operator	NOUN
ejpam-4412	410	25	,	,	PUNCT
ejpam-4412	410	26	we	we	PRON
ejpam-4412	410	27	have	have	VERB
ejpam-4412	410	28	t2|ran(w	t2|ran(w	ADV
ejpam-4412	410	29	)	)	PUNCT
ejpam-4412	411	1	=	=	SYM
ejpam-4412	411	2	t2(s	t2(s	PROPN
ejpam-4412	411	3	,	,	PUNCT
ejpam-4412	411	4	t)|ran(w	t)|ran(w	NUM
ejpam-4412	411	5	)	)	PUNCT
ejpam-4412	411	6	and	and	CCONJ
ejpam-4412	411	7	s∗	s∗	PROPN
ejpam-4412	411	8	2	2	NUM
ejpam-4412	411	9	|ker(w	|ker(w	NOUN
ejpam-4412	411	10	)	)	PUNCT
ejpam-4412	412	1	⊥	⊥	NOUN
ejpam-4412	412	2	=	=	PUNCT
ejpam-4412	412	3	s∗	s∗	PROPN
ejpam-4412	412	4	2(s	2(s	NUM
ejpam-4412	412	5	,	,	PUNCT
ejpam-4412	412	6	t)|ker(w	t)|ker(w	PRON
ejpam-4412	412	7	)	)	PUNCT
ejpam-4412	413	1	⊥	⊥	NOUN
ejpam-4412	413	2	by	by	ADP
ejpam-4412	413	3	lemma	lemma	PROPN
ejpam-4412	413	4	9	9	NUM
ejpam-4412	413	5	.	.	PUNCT
ejpam-4412	413	6	since	since	SCONJ
ejpam-4412	413	7	t2	t2	NOUN
ejpam-4412	413	8	,	,	PUNCT
ejpam-4412	413	9	s	s	PART
ejpam-4412	413	10	∗	∗	NOUN
ejpam-4412	413	11	2	2	NUM
ejpam-4412	413	12	are	be	AUX
ejpam-4412	413	13	pure	pure	ADJ
ejpam-4412	413	14	,	,	PUNCT
ejpam-4412	413	15	it	it	PRON
ejpam-4412	413	16	implies	imply	VERB
ejpam-4412	413	17	w	w	NOUN
ejpam-4412	413	18	=	=	SYM
ejpam-4412	413	19	|t2|sx22|s∗	|t2|sx22|s∗	NUM
ejpam-4412	413	20	2	2	NUM
ejpam-4412	413	21	|s	|s	X
ejpam-4412	413	22	=	=	SYM
ejpam-4412	413	23	0	0	X
ejpam-4412	413	24	.	.	PUNCT
ejpam-4412	414	1	hence	hence	ADV
ejpam-4412	414	2	x22	x22	NUM
ejpam-4412	414	3	=	=	SYM
ejpam-4412	414	4	0	0	X
ejpam-4412	414	5	.	.	PUNCT
ejpam-4412	415	1	similarly	similarly	ADV
ejpam-4412	415	2	x12	x12	NUM
ejpam-4412	415	3	=	=	SYM
ejpam-4412	415	4	0	0	NUM
ejpam-4412	415	5	,	,	PUNCT
ejpam-4412	415	6	x21	x21	PROPN
ejpam-4412	416	1	=	=	SYM
ejpam-4412	416	2	0	0	X
ejpam-4412	416	3	.	.	PUNCT
ejpam-4412	417	1	hence	hence	ADV
ejpam-4412	417	2	x	x	X
ejpam-4412	417	3	=	=	PUNCT
ejpam-4412	417	4	x11	x11	PROPN
ejpam-4412	417	5	and	and	CCONJ
ejpam-4412	417	6	s	s	PROPN
ejpam-4412	417	7	,	,	PUNCT
ejpam-4412	417	8	t	t	PROPN
ejpam-4412	417	9	are	be	AUX
ejpam-4412	417	10	unitarily	unitarily	ADV
ejpam-4412	417	11	equivalent	equivalent	ADJ
ejpam-4412	417	12	normal	normal	ADJ
ejpam-4412	417	13	operators	operator	NOUN
ejpam-4412	417	14	.	.	PUNCT
ejpam-4412	418	1	the	the	DET
ejpam-4412	418	2	following	follow	VERB
ejpam-4412	418	3	lemma	lemma	PROPN
ejpam-4412	418	4	is	be	AUX
ejpam-4412	418	5	due	due	ADJ
ejpam-4412	418	6	to	to	ADP
ejpam-4412	418	7	williams	williams	PROPN
ejpam-4412	418	8	[	[	X
ejpam-4412	418	9	32	32	NUM
ejpam-4412	418	10	,	,	PUNCT
ejpam-4412	418	11	lemma	lemma	PROPN
ejpam-4412	418	12	1.1	1.1	NUM
ejpam-4412	418	13	]	]	PUNCT
ejpam-4412	418	14	lemma	lemma	PROPN
ejpam-4412	418	15	13	13	NUM
ejpam-4412	418	16	.	.	PUNCT
ejpam-4412	419	1	let	let	VERB
ejpam-4412	419	2	n1	n1	PROPN
ejpam-4412	419	3	∈	∈	PROPN
ejpam-4412	419	4	b(h	b(h	PROPN
ejpam-4412	419	5	)	)	PUNCT
ejpam-4412	419	6	and	and	CCONJ
ejpam-4412	419	7	n2	n2	PROPN
ejpam-4412	419	8	∈	∈	PROPN
ejpam-4412	419	9	b(k	b(k	PROPN
ejpam-4412	419	10	)	)	PUNCT
ejpam-4412	419	11	be	be	AUX
ejpam-4412	419	12	normal	normal	ADJ
ejpam-4412	419	13	.	.	PUNCT
ejpam-4412	420	1	if	if	SCONJ
ejpam-4412	420	2	x	x	SYM
ejpam-4412	420	3	∈	∈	PROPN
ejpam-4412	420	4	b(k	b(k	PROPN
ejpam-4412	420	5	,	,	PUNCT
ejpam-4412	420	6	h	h	NOUN
ejpam-4412	420	7	)	)	PUNCT
ejpam-4412	420	8	and	and	CCONJ
ejpam-4412	420	9	y	y	PROPN
ejpam-4412	420	10	∈	∈	PROPN
ejpam-4412	420	11	b(h	b(h	PROPN
ejpam-4412	420	12	,	,	PUNCT
ejpam-4412	420	13	k	k	NOUN
ejpam-4412	420	14	)	)	PUNCT
ejpam-4412	420	15	are	be	AUX
ejpam-4412	420	16	injective	injective	ADJ
ejpam-4412	420	17	such	such	ADJ
ejpam-4412	420	18	that	that	DET
ejpam-4412	420	19	n1x	n1x	PROPN
ejpam-4412	420	20	=	=	PUNCT
ejpam-4412	420	21	xn2	xn2	PROPN
ejpam-4412	420	22	and	and	CCONJ
ejpam-4412	420	23	y	y	PROPN
ejpam-4412	420	24	n1	n1	PROPN
ejpam-4412	420	25	=	=	SYM
ejpam-4412	420	26	n2y	n2y	NOUN
ejpam-4412	420	27	,	,	PUNCT
ejpam-4412	420	28	then	then	ADV
ejpam-4412	420	29	n1	n1	PROPN
ejpam-4412	420	30	and	and	CCONJ
ejpam-4412	420	31	n2	n2	NOUN
ejpam-4412	420	32	are	be	AUX
ejpam-4412	420	33	unitarily	unitarily	ADV
ejpam-4412	420	34	equivalent	equivalent	ADJ
ejpam-4412	420	35	.	.	PUNCT
ejpam-4412	421	1	stampfli	stampfli	NOUN
ejpam-4412	421	2	and	and	CCONJ
ejpam-4412	421	3	wadhwa	wadhwa	ADJ
ejpam-4412	422	1	[	[	X
ejpam-4412	422	2	28	28	NUM
ejpam-4412	422	3	]	]	PUNCT
ejpam-4412	422	4	proved	prove	VERB
ejpam-4412	422	5	that	that	SCONJ
ejpam-4412	422	6	the	the	DET
ejpam-4412	422	7	normal	normal	ADJ
ejpam-4412	422	8	parts	part	NOUN
ejpam-4412	422	9	of	of	ADP
ejpam-4412	422	10	quasisimilar	quasisimilar	ADJ
ejpam-4412	422	11	dominant	dominant	ADJ
ejpam-4412	422	12	operators	operator	NOUN
ejpam-4412	422	13	are	be	AUX
ejpam-4412	422	14	unitarily	unitarily	ADV
ejpam-4412	422	15	equivalent	equivalent	ADJ
ejpam-4412	422	16	.	.	PUNCT
ejpam-4412	423	1	this	this	DET
ejpam-4412	423	2	result	result	NOUN
ejpam-4412	423	3	was	be	AUX
ejpam-4412	423	4	generalized	generalize	VERB
ejpam-4412	423	5	to	to	ADP
ejpam-4412	423	6	classes	class	NOUN
ejpam-4412	423	7	of	of	ADP
ejpam-4412	423	8	p	p	NOUN
ejpam-4412	423	9	-	-	PUNCT
ejpam-4412	423	10	hyponormal	hyponormal	ADJ
ejpam-4412	423	11	operators	operator	NOUN
ejpam-4412	423	12	in	in	ADP
ejpam-4412	423	13	[	[	X
ejpam-4412	423	14	12	12	NUM
ejpam-4412	423	15	]	]	PUNCT
ejpam-4412	423	16	.	.	PUNCT
ejpam-4412	424	1	we	we	PRON
ejpam-4412	424	2	prove	prove	VERB
ejpam-4412	424	3	that	that	SCONJ
ejpam-4412	424	4	theses	theses	PROPN
ejpam-4412	424	5	results	result	NOUN
ejpam-4412	424	6	hold	hold	VERB
ejpam-4412	424	7	for	for	ADP
ejpam-4412	424	8	class	class	NOUN
ejpam-4412	424	9	p	p	NOUN
ejpam-4412	424	10	-	-	PUNCT
ejpam-4412	424	11	wa(s	wa(s	NUM
ejpam-4412	424	12	,	,	PUNCT
ejpam-4412	424	13	t	t	NOUN
ejpam-4412	424	14	)	)	PUNCT
ejpam-4412	424	15	operators	operator	NOUN
ejpam-4412	424	16	.	.	PUNCT
ejpam-4412	425	1	theorem	theorem	NOUN
ejpam-4412	425	2	9	9	NUM
ejpam-4412	425	3	.	.	PUNCT
ejpam-4412	425	4	suppose	suppose	VERB
ejpam-4412	425	5	that	that	SCONJ
ejpam-4412	425	6	0	0	NUM
ejpam-4412	425	7	<	<	X
ejpam-4412	425	8	s	s	PROPN
ejpam-4412	425	9	,	,	PUNCT
ejpam-4412	425	10	t	t	PROPN
ejpam-4412	425	11	,	,	PUNCT
ejpam-4412	425	12	s	s	PART
ejpam-4412	425	13	+	+	NUM
ejpam-4412	425	14	t	t	X
ejpam-4412	425	15	=	=	SYM
ejpam-4412	425	16	1	1	NUM
ejpam-4412	425	17	and	and	CCONJ
ejpam-4412	425	18	)	)	PUNCT
ejpam-4412	425	19	<	<	X
ejpam-4412	426	1	p	p	X
ejpam-4412	426	2	≤	≤	NUM
ejpam-4412	426	3	1	1	NUM
ejpam-4412	426	4	.	.	PUNCT
ejpam-4412	427	1	for	for	ADP
ejpam-4412	427	2	each	each	DET
ejpam-4412	427	3	i	i	NOUN
ejpam-4412	427	4	=	=	NOUN
ejpam-4412	427	5	1	1	NUM
ejpam-4412	427	6	,	,	PUNCT
ejpam-4412	427	7	2	2	NUM
ejpam-4412	427	8	,	,	PUNCT
ejpam-4412	427	9	let	let	VERB
ejpam-4412	427	10	ti	ti	PRON
ejpam-4412	427	11	∈	∈	PROPN
ejpam-4412	427	12	b(hi	b(hi	PROPN
ejpam-4412	427	13	)	)	PUNCT
ejpam-4412	427	14	be	be	AUX
ejpam-4412	427	15	class	class	NOUN
ejpam-4412	427	16	p	p	NOUN
ejpam-4412	427	17	-	-	PUNCT
ejpam-4412	427	18	wa(s	wa(s	NUM
ejpam-4412	427	19	,	,	PUNCT
ejpam-4412	427	20	t	t	PROPN
ejpam-4412	427	21	)	)	PUNCT
ejpam-4412	427	22	operators	operator	NOUN
ejpam-4412	427	23	such	such	ADJ
ejpam-4412	427	24	that	that	DET
ejpam-4412	427	25	ker(tj	ker(tj	NOUN
ejpam-4412	427	26	)	)	PUNCT
ejpam-4412	428	1	⊂	⊂	PROPN
ejpam-4412	428	2	ker(t	ker(t	PROPN
ejpam-4412	428	3	∗	∗	X
ejpam-4412	428	4	j	j	PROPN
ejpam-4412	428	5	)	)	PUNCT
ejpam-4412	429	1	and	and	CCONJ
ejpam-4412	429	2	let	let	VERB
ejpam-4412	429	3	ti	ti	PROPN
ejpam-4412	429	4	=	=	SYM
ejpam-4412	429	5	ni	ni	PROPN
ejpam-4412	429	6	⊕	⊕	PROPN
ejpam-4412	429	7	vi	vi	PROPN
ejpam-4412	429	8	on	on	ADP
ejpam-4412	429	9	hi	hi	ADV
ejpam-4412	429	10	=	=	PUNCT
ejpam-4412	429	11	hi1	hi1	ADP
ejpam-4412	429	12	⊕	⊕	PROPN
ejpam-4412	429	13	hi2	hi2	PROPN
ejpam-4412	429	14	,	,	PUNCT
ejpam-4412	429	15	where	where	SCONJ
ejpam-4412	429	16	ni	ni	PROPN
ejpam-4412	429	17	and	and	CCONJ
ejpam-4412	429	18	vi	vi	PROPN
ejpam-4412	429	19	are	be	AUX
ejpam-4412	429	20	the	the	DET
ejpam-4412	429	21	normal	normal	ADJ
ejpam-4412	429	22	and	and	CCONJ
ejpam-4412	429	23	pure	pure	ADJ
ejpam-4412	429	24	parts	part	NOUN
ejpam-4412	429	25	,	,	PUNCT
ejpam-4412	429	26	respectively	respectively	ADV
ejpam-4412	429	27	of	of	ADP
ejpam-4412	429	28	ti	ti	NOUN
ejpam-4412	429	29	.	.	PUNCT
ejpam-4412	430	1	if	if	SCONJ
ejpam-4412	430	2	t1	t1	NOUN
ejpam-4412	430	3	and	and	CCONJ
ejpam-4412	430	4	t2	t2	NOUN
ejpam-4412	430	5	are	be	AUX
ejpam-4412	430	6	quasisimilar	quasisimilar	ADJ
ejpam-4412	430	7	,	,	PUNCT
ejpam-4412	430	8	then	then	ADV
ejpam-4412	430	9	n1	n1	PROPN
ejpam-4412	430	10	and	and	CCONJ
ejpam-4412	430	11	n2	n2	NOUN
ejpam-4412	430	12	are	be	AUX
ejpam-4412	430	13	unitarily	unitarily	ADV
ejpam-4412	430	14	equivalent	equivalent	ADJ
ejpam-4412	430	15	and	and	CCONJ
ejpam-4412	430	16	there	there	PRON
ejpam-4412	430	17	exist	exist	VERB
ejpam-4412	430	18	x∗	x∗	PROPN
ejpam-4412	430	19	∈	∈	PROPN
ejpam-4412	430	20	b(h22,h12	b(h22,h12	PROPN
ejpam-4412	430	21	)	)	PUNCT
ejpam-4412	430	22	and	and	CCONJ
ejpam-4412	430	23	y∗	y∗	PROPN
ejpam-4412	430	24	∈	∈	PROPN
ejpam-4412	430	25	b(h12,h22	b(h12,h22	NOUN
ejpam-4412	430	26	)	)	PUNCT
ejpam-4412	430	27	having	have	VERB
ejpam-4412	430	28	dense	dense	ADJ
ejpam-4412	430	29	range	range	NOUN
ejpam-4412	430	30	such	such	ADJ
ejpam-4412	430	31	that	that	DET
ejpam-4412	430	32	v1x∗	v1x∗	PROPN
ejpam-4412	430	33	=	=	SYM
ejpam-4412	430	34	x∗v2	x∗v2	PROPN
ejpam-4412	430	35	and	and	CCONJ
ejpam-4412	430	36	y∗v1	y∗v1	NOUN
ejpam-4412	430	37	=	=	PRON
ejpam-4412	431	1	v2y∗.	v2y∗.	PRON
ejpam-4412	431	2	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	431	3	,	,	PUNCT
ejpam-4412	431	4	n.	n.	NOUN
ejpam-4412	431	5	h.	h.	PROPN
ejpam-4412	431	6	altaweel	altaweel	PROPN
ejpam-4412	431	7	/	/	SYM
ejpam-4412	431	8	eur	eur	PROPN
ejpam-4412	431	9	.	.	PUNCT
ejpam-4412	432	1	j.	j.	PROPN
ejpam-4412	432	2	pure	pure	PROPN
ejpam-4412	432	3	appl	appl	PROPN
ejpam-4412	432	4	.	.	PROPN
ejpam-4412	432	5	math	math	PROPN
ejpam-4412	432	6	,	,	PUNCT
ejpam-4412	432	7	15	15	NUM
ejpam-4412	432	8	(	(	PUNCT
ejpam-4412	432	9	3	3	NUM
ejpam-4412	432	10	)	)	PUNCT
ejpam-4412	432	11	(	(	PUNCT
ejpam-4412	432	12	2022	2022	NUM
ejpam-4412	432	13	)	)	PUNCT
ejpam-4412	432	14	,	,	PUNCT
ejpam-4412	432	15	1067	1067	NUM
ejpam-4412	432	16	-	-	SYM
ejpam-4412	432	17	1089	1089	NUM
ejpam-4412	432	18	1081	1081	NUM
ejpam-4412	432	19	proof	proof	NOUN
ejpam-4412	432	20	.	.	PUNCT
ejpam-4412	433	1	by	by	ADP
ejpam-4412	433	2	hypothesis	hypothesis	NOUN
ejpam-4412	433	3	there	there	PRON
ejpam-4412	433	4	exist	exist	VERB
ejpam-4412	433	5	quasiaffinities	quasiaffinitie	NOUN
ejpam-4412	433	6	x	x	SYM
ejpam-4412	433	7	∈	∈	NOUN
ejpam-4412	433	8	b(h2,h1	b(h2,h1	NOUN
ejpam-4412	433	9	)	)	PUNCT
ejpam-4412	433	10	and	and	CCONJ
ejpam-4412	433	11	y	y	PROPN
ejpam-4412	433	12	∈	∈	PROPN
ejpam-4412	433	13	b(h1,h2	b(h1,h2	PROPN
ejpam-4412	433	14	)	)	PUNCT
ejpam-4412	433	15	such	such	ADJ
ejpam-4412	433	16	that	that	SCONJ
ejpam-4412	433	17	t1x	t1x	PROPN
ejpam-4412	433	18	=	=	SYM
ejpam-4412	433	19	xt2	xt2	PROPN
ejpam-4412	433	20	and	and	CCONJ
ejpam-4412	433	21	y	y	PROPN
ejpam-4412	433	22	t1	t1	NOUN
ejpam-4412	433	23	=	=	PUNCT
ejpam-4412	433	24	t2y	t2y	NOUN
ejpam-4412	433	25	.	.	PUNCT
ejpam-4412	434	1	let	let	VERB
ejpam-4412	434	2	x	x	PUNCT
ejpam-4412	434	3	=	=	PRON
ejpam-4412	434	4	(	(	PUNCT
ejpam-4412	434	5	x1	x1	PROPN
ejpam-4412	434	6	x2	x2	PROPN
ejpam-4412	434	7	x3	x3	PROPN
ejpam-4412	434	8	x4	x4	PROPN
ejpam-4412	434	9	)	)	PUNCT
ejpam-4412	434	10	and	and	CCONJ
ejpam-4412	434	11	y	y	PROPN
ejpam-4412	435	1	=	=	PUNCT
ejpam-4412	435	2	(	(	PUNCT
ejpam-4412	435	3	y1	y1	INTJ
ejpam-4412	435	4	y2	y2	NOUN
ejpam-4412	435	5	y3	y3	NOUN
ejpam-4412	435	6	y4	y4	PROPN
ejpam-4412	435	7	)	)	PUNCT
ejpam-4412	435	8	with	with	ADP
ejpam-4412	435	9	respect	respect	NOUN
ejpam-4412	435	10	to	to	ADP
ejpam-4412	435	11	h2	h2	NOUN
ejpam-4412	435	12	=	=	SYM
ejpam-4412	435	13	h21	h21	PROPN
ejpam-4412	435	14	⊕	⊕	PROPN
ejpam-4412	435	15	h22	h22	PROPN
ejpam-4412	435	16	and	and	CCONJ
ejpam-4412	435	17	h1	h1	PROPN
ejpam-4412	435	18	=	=	PROPN
ejpam-4412	435	19	h11	h11	PROPN
ejpam-4412	435	20	⊕	⊕	PROPN
ejpam-4412	435	21	h12	h12	PROPN
ejpam-4412	435	22	,	,	PUNCT
ejpam-4412	435	23	respectively	respectively	ADV
ejpam-4412	435	24	.	.	PUNCT
ejpam-4412	436	1	a	a	DET
ejpam-4412	436	2	simple	simple	ADJ
ejpam-4412	436	3	matrix	matrix	NOUN
ejpam-4412	436	4	calculation	calculation	NOUN
ejpam-4412	436	5	shows	show	VERB
ejpam-4412	436	6	that	that	SCONJ
ejpam-4412	436	7	v1x3	v1x3	PROPN
ejpam-4412	436	8	=	=	SYM
ejpam-4412	436	9	x3n2	x3n2	PROPN
ejpam-4412	436	10	and	and	CCONJ
ejpam-4412	436	11	v2y3	v2y3	X
ejpam-4412	437	1	=	=	X
ejpam-4412	437	2	y3n1	y3n1	INTJ
ejpam-4412	437	3	.	.	PUNCT
ejpam-4412	438	1	we	we	PRON
ejpam-4412	438	2	claim	claim	VERB
ejpam-4412	438	3	that	that	SCONJ
ejpam-4412	438	4	x3	x3	ADJ
ejpam-4412	438	5	=	=	SYM
ejpam-4412	438	6	y3	y3	NOUN
ejpam-4412	438	7	=	=	SYM
ejpam-4412	438	8	0	0	X
ejpam-4412	438	9	.	.	PUNCT
ejpam-4412	439	1	let	let	VERB
ejpam-4412	439	2	m	m	VERB
ejpam-4412	439	3	=	=	PUNCT
ejpam-4412	439	4	ran(x3	ran(x3	NOUN
ejpam-4412	439	5	)	)	PUNCT
ejpam-4412	439	6	.	.	PUNCT
ejpam-4412	440	1	then	then	ADV
ejpam-4412	440	2	m	m	PROPN
ejpam-4412	440	3	is	be	AUX
ejpam-4412	440	4	a	a	DET
ejpam-4412	440	5	non	non	ADJ
ejpam-4412	440	6	-	-	ADJ
ejpam-4412	440	7	trivial	trivial	ADJ
ejpam-4412	440	8	invariant	invariant	ADJ
ejpam-4412	440	9	subspace	subspace	NOUN
ejpam-4412	440	10	of	of	ADP
ejpam-4412	440	11	v1	v1	NOUN
ejpam-4412	440	12	.	.	PUNCT
ejpam-4412	441	1	since	since	SCONJ
ejpam-4412	441	2	v	v	NUM
ejpam-4412	441	3	∗	∗	NOUN
ejpam-4412	441	4	1	1	NUM
ejpam-4412	441	5	x3	x3	NOUN
ejpam-4412	441	6	=	=	SYM
ejpam-4412	441	7	x3n	x3n	PROPN
ejpam-4412	441	8	∗	∗	NOUN
ejpam-4412	441	9	2	2	NUM
ejpam-4412	441	10	by	by	ADP
ejpam-4412	441	11	proposition	proposition	NOUN
ejpam-4412	441	12	2	2	NUM
ejpam-4412	441	13	,	,	PUNCT
ejpam-4412	441	14	m	m	VERB
ejpam-4412	441	15	is	be	AUX
ejpam-4412	441	16	an	an	DET
ejpam-4412	441	17	invariant	invariant	ADJ
ejpam-4412	441	18	subspace	subspace	NOUN
ejpam-4412	441	19	of	of	ADP
ejpam-4412	441	20	v	v	NUM
ejpam-4412	441	21	∗	∗	NOUN
ejpam-4412	441	22	1	1	NUM
ejpam-4412	441	23	.	.	PUNCT
ejpam-4412	442	1	hence	hence	ADV
ejpam-4412	442	2	m	m	VERB
ejpam-4412	442	3	reduces	reduce	VERB
ejpam-4412	442	4	v1	v1	NOUN
ejpam-4412	442	5	,	,	PUNCT
ejpam-4412	442	6	σ(v1|m	σ(v1|m	PROPN
ejpam-4412	442	7	)	)	PUNCT
ejpam-4412	442	8	⊂	⊂	PROPN
ejpam-4412	442	9	σ(v1	σ(v1	X
ejpam-4412	442	10	)	)	PUNCT
ejpam-4412	442	11	and	and	CCONJ
ejpam-4412	442	12	v1|m	v1|m	NOUN
ejpam-4412	442	13	is	be	AUX
ejpam-4412	442	14	invertible	invertible	ADJ
ejpam-4412	442	15	.	.	PUNCT
ejpam-4412	443	1	let	let	VERB
ejpam-4412	443	2	v	v	NOUN
ejpam-4412	443	3	′	′	NOUN
ejpam-4412	443	4	1	1	NUM
ejpam-4412	443	5	=	=	NOUN
ejpam-4412	443	6	v1|m	v1|m	NOUN
ejpam-4412	443	7	and	and	CCONJ
ejpam-4412	443	8	define	define	VERB
ejpam-4412	443	9	an	an	DET
ejpam-4412	443	10	operator	operator	NOUN
ejpam-4412	443	11	x	x	PUNCT
ejpam-4412	443	12	′	′	NOUN
ejpam-4412	443	13	3	3	NUM
ejpam-4412	443	14	:	:	PUNCT
ejpam-4412	443	15	h12	h12	NOUN
ejpam-4412	443	16	−→	−→	NOUN
ejpam-4412	443	17	m	m	VERB
ejpam-4412	443	18	by	by	ADP
ejpam-4412	443	19	x	x	X
ejpam-4412	443	20	′	′	NUM
ejpam-4412	443	21	3x	3x	NUM
ejpam-4412	443	22	=	=	PUNCT
ejpam-4412	444	1	x3x	x3x	PROPN
ejpam-4412	444	2	for	for	ADP
ejpam-4412	444	3	each	each	DET
ejpam-4412	444	4	x	x	SYM
ejpam-4412	444	5	∈	∈	PROPN
ejpam-4412	444	6	h12	h12	NOUN
ejpam-4412	444	7	.	.	PUNCT
ejpam-4412	445	1	then	then	ADV
ejpam-4412	445	2	v	v	NOUN
ejpam-4412	445	3	′	′	NUM
ejpam-4412	445	4	1	1	NUM
ejpam-4412	445	5	is	be	AUX
ejpam-4412	445	6	class	class	NOUN
ejpam-4412	445	7	p	p	NOUN
ejpam-4412	445	8	-	-	PUNCT
ejpam-4412	445	9	wa(s	wa(s	NUM
ejpam-4412	445	10	,	,	PUNCT
ejpam-4412	445	11	t	t	PROPN
ejpam-4412	445	12	)	)	PUNCT
ejpam-4412	445	13	by	by	ADP
ejpam-4412	445	14	lemma	lemma	PROPN
ejpam-4412	445	15	6	6	NUM
ejpam-4412	445	16	,	,	PUNCT
ejpam-4412	445	17	so	so	SCONJ
ejpam-4412	445	18	that	that	SCONJ
ejpam-4412	445	19	x	x	X
ejpam-4412	445	20	′	′	NOUN
ejpam-4412	445	21	3	3	NUM
ejpam-4412	445	22	has	have	VERB
ejpam-4412	445	23	dense	dense	ADJ
ejpam-4412	445	24	range	range	NOUN
ejpam-4412	445	25	and	and	CCONJ
ejpam-4412	445	26	satisfies	satisfie	NOUN
ejpam-4412	445	27	v	v	ADP
ejpam-4412	445	28	′	′	NUM
ejpam-4412	445	29	1x	1x	NUM
ejpam-4412	446	1	′	′	NOUN
ejpam-4412	446	2	3	3	NUM
ejpam-4412	447	1	=	=	NOUN
ejpam-4412	447	2	x	x	SYM
ejpam-4412	447	3	′	′	NUM
ejpam-4412	447	4	3n2	3n2	NUM
ejpam-4412	447	5	.	.	PUNCT
ejpam-4412	448	1	hence	hence	ADV
ejpam-4412	448	2	v	v	NOUN
ejpam-4412	448	3	′	′	NUM
ejpam-4412	448	4	1	1	NUM
ejpam-4412	448	5	is	be	AUX
ejpam-4412	448	6	normal	normal	ADJ
ejpam-4412	448	7	by	by	ADP
ejpam-4412	448	8	propsition	propsition	NOUN
ejpam-4412	448	9	2	2	NUM
ejpam-4412	448	10	.	.	PUNCT
ejpam-4412	449	1	since	since	SCONJ
ejpam-4412	449	2	v1	v1	NOUN
ejpam-4412	449	3	is	be	AUX
ejpam-4412	449	4	pure	pure	ADJ
ejpam-4412	449	5	,	,	PUNCT
ejpam-4412	449	6	this	this	PRON
ejpam-4412	449	7	implies	imply	VERB
ejpam-4412	449	8	that	that	SCONJ
ejpam-4412	449	9	m	m	VERB
ejpam-4412	449	10	=	=	SYM
ejpam-4412	449	11	{	{	PUNCT
ejpam-4412	449	12	0	0	NUM
ejpam-4412	449	13	}	}	PUNCT
ejpam-4412	449	14	and	and	CCONJ
ejpam-4412	449	15	x3	x3	ADJ
ejpam-4412	449	16	=	=	SYM
ejpam-4412	449	17	0	0	X
ejpam-4412	449	18	.	.	PUNCT
ejpam-4412	450	1	similarly	similarly	ADV
ejpam-4412	450	2	,	,	PUNCT
ejpam-4412	450	3	we	we	PRON
ejpam-4412	450	4	have	have	VERB
ejpam-4412	450	5	y3	y3	NOUN
ejpam-4412	450	6	=	=	SYM
ejpam-4412	450	7	0	0	NUM
ejpam-4412	450	8	.	.	PUNCT
ejpam-4412	451	1	hence	hence	ADV
ejpam-4412	451	2	x1	x1	PROPN
ejpam-4412	451	3	and	and	CCONJ
ejpam-4412	451	4	y1	y1	NOUN
ejpam-4412	451	5	are	be	AUX
ejpam-4412	451	6	injective	injective	ADJ
ejpam-4412	451	7	.	.	PUNCT
ejpam-4412	452	1	since	since	SCONJ
ejpam-4412	452	2	n1x1	n1x1	PROPN
ejpam-4412	452	3	=	=	SYM
ejpam-4412	452	4	x1n2	x1n2	PROPN
ejpam-4412	452	5	and	and	CCONJ
ejpam-4412	452	6	y1n1	y1n1	PROPN
ejpam-4412	452	7	=	=	SYM
ejpam-4412	452	8	n2y1	n2y1	PROPN
ejpam-4412	452	9	,	,	PUNCT
ejpam-4412	452	10	n1	n1	PROPN
ejpam-4412	452	11	and	and	CCONJ
ejpam-4412	452	12	n2	n2	NOUN
ejpam-4412	452	13	are	be	AUX
ejpam-4412	452	14	unitarily	unitarily	ADV
ejpam-4412	452	15	equivalent	equivalent	ADJ
ejpam-4412	452	16	,	,	PUNCT
ejpam-4412	452	17	by	by	ADP
ejpam-4412	452	18	lemma	lemma	PROPN
ejpam-4412	452	19	13	13	NUM
ejpam-4412	452	20	.	.	PUNCT
ejpam-4412	453	1	also	also	ADV
ejpam-4412	453	2	,	,	PUNCT
ejpam-4412	453	3	x4	x4	PROPN
ejpam-4412	453	4	and	and	CCONJ
ejpam-4412	453	5	y4	y4	PROPN
ejpam-4412	453	6	have	have	VERB
ejpam-4412	453	7	dense	dense	ADJ
ejpam-4412	453	8	ranges	range	NOUN
ejpam-4412	453	9	.	.	PUNCT
ejpam-4412	454	1	hence	hence	ADV
ejpam-4412	454	2	v1x4	v1x4	VERB
ejpam-4412	454	3	=	=	PUNCT
ejpam-4412	454	4	x4v2	x4v2	PROPN
ejpam-4412	454	5	and	and	CCONJ
ejpam-4412	454	6	y4v1	y4v1	NOUN
ejpam-4412	454	7	=	=	SYM
ejpam-4412	454	8	v2y4	v2y4	PROPN
ejpam-4412	454	9	,	,	PUNCT
ejpam-4412	454	10	so	so	SCONJ
ejpam-4412	454	11	the	the	DET
ejpam-4412	454	12	proof	proof	NOUN
ejpam-4412	454	13	is	be	AUX
ejpam-4412	454	14	complete	complete	ADJ
ejpam-4412	454	15	.	.	PUNCT
ejpam-4412	455	1	corollary	corollary	ADJ
ejpam-4412	455	2	3	3	X
ejpam-4412	455	3	.	.	PUNCT
ejpam-4412	456	1	let	let	VERB
ejpam-4412	456	2	t1	t1	PROPN
ejpam-4412	456	3	∈	∈	PROPN
ejpam-4412	456	4	b(h1	b(h1	NOUN
ejpam-4412	456	5	)	)	PUNCT
ejpam-4412	456	6	and	and	CCONJ
ejpam-4412	456	7	t2	t2	PROPN
ejpam-4412	456	8	∈	∈	PROPN
ejpam-4412	456	9	b(h2	b(h2	NOUN
ejpam-4412	456	10	)	)	PUNCT
ejpam-4412	456	11	be	be	AUX
ejpam-4412	456	12	quasisimilar	quasisimilar	ADJ
ejpam-4412	456	13	class	class	NOUN
ejpam-4412	456	14	p	p	NOUN
ejpam-4412	456	15	-	-	PUNCT
ejpam-4412	456	16	wa(s	wa(s	NUM
ejpam-4412	456	17	,	,	PUNCT
ejpam-4412	456	18	t	t	NOUN
ejpam-4412	456	19	)	)	PUNCT
ejpam-4412	456	20	operators	operator	NOUN
ejpam-4412	456	21	for	for	ADP
ejpam-4412	456	22	0	0	NUM
ejpam-4412	456	23	<	<	X
ejpam-4412	456	24	s	s	PROPN
ejpam-4412	456	25	,	,	PUNCT
ejpam-4412	456	26	t	t	PROPN
ejpam-4412	456	27	,	,	PUNCT
ejpam-4412	456	28	s+	s+	ADP
ejpam-4412	456	29	t	t	PROPN
ejpam-4412	456	30	=	=	SYM
ejpam-4412	456	31	1	1	NUM
ejpam-4412	456	32	and	and	CCONJ
ejpam-4412	456	33	0	0	NUM
ejpam-4412	456	34	<	<	X
ejpam-4412	456	35	p	p	X
ejpam-4412	456	36	≤	≤	NUM
ejpam-4412	456	37	1	1	NUM
ejpam-4412	456	38	.	.	PUNCT
ejpam-4412	457	1	if	if	SCONJ
ejpam-4412	457	2	t1	t1	NOUN
ejpam-4412	457	3	is	be	AUX
ejpam-4412	457	4	pure	pure	ADJ
ejpam-4412	457	5	,	,	PUNCT
ejpam-4412	457	6	then	then	ADV
ejpam-4412	457	7	t2	t2	NOUN
ejpam-4412	457	8	is	be	AUX
ejpam-4412	457	9	also	also	ADV
ejpam-4412	457	10	pure	pure	ADJ
ejpam-4412	457	11	.	.	PUNCT
ejpam-4412	458	1	corollary	corollary	ADJ
ejpam-4412	458	2	4	4	NUM
ejpam-4412	458	3	.	.	PUNCT
ejpam-4412	459	1	let	let	VERB
ejpam-4412	459	2	t1	t1	PROPN
ejpam-4412	459	3	∈	∈	PROPN
ejpam-4412	459	4	b(h1	b(h1	NOUN
ejpam-4412	459	5	)	)	PUNCT
ejpam-4412	459	6	be	be	AUX
ejpam-4412	459	7	class	class	NOUN
ejpam-4412	459	8	p	p	NOUN
ejpam-4412	459	9	-	-	PUNCT
ejpam-4412	459	10	wa(s	wa(s	NUM
ejpam-4412	459	11	,	,	PUNCT
ejpam-4412	459	12	t	t	NOUN
ejpam-4412	459	13	)	)	PUNCT
ejpam-4412	459	14	operators	operator	NOUN
ejpam-4412	459	15	for	for	ADP
ejpam-4412	459	16	0	0	NUM
ejpam-4412	459	17	<	<	X
ejpam-4412	459	18	s	s	PROPN
ejpam-4412	459	19	,	,	PUNCT
ejpam-4412	459	20	t	t	PROPN
ejpam-4412	459	21	,	,	PUNCT
ejpam-4412	459	22	s	s	PART
ejpam-4412	460	1	+	+	NUM
ejpam-4412	460	2	t	t	X
ejpam-4412	460	3	=	=	SYM
ejpam-4412	460	4	1	1	NUM
ejpam-4412	460	5	and	and	CCONJ
ejpam-4412	460	6	0	0	NUM
ejpam-4412	460	7	<	<	X
ejpam-4412	460	8	p	p	X
ejpam-4412	460	9	≤	≤	NUM
ejpam-4412	460	10	1	1	NUM
ejpam-4412	460	11	and	and	CCONJ
ejpam-4412	460	12	t2	t2	PROPN
ejpam-4412	460	13	∈	∈	PROPN
ejpam-4412	460	14	b(h2	b(h2	NOUN
ejpam-4412	460	15	)	)	PUNCT
ejpam-4412	460	16	be	be	AUX
ejpam-4412	460	17	normal	normal	ADJ
ejpam-4412	460	18	.	.	PUNCT
ejpam-4412	461	1	if	if	SCONJ
ejpam-4412	461	2	t1	t1	NOUN
ejpam-4412	461	3	and	and	CCONJ
ejpam-4412	461	4	t2	t2	NOUN
ejpam-4412	461	5	are	be	AUX
ejpam-4412	461	6	quasisimilar	quasisimilar	ADJ
ejpam-4412	461	7	,	,	PUNCT
ejpam-4412	461	8	then	then	ADV
ejpam-4412	461	9	t1	t1	NOUN
ejpam-4412	461	10	and	and	CCONJ
ejpam-4412	461	11	t2	t2	NOUN
ejpam-4412	461	12	are	be	AUX
ejpam-4412	461	13	unitarily	unitarily	ADV
ejpam-4412	461	14	equivalent	equivalent	ADJ
ejpam-4412	461	15	normal	normal	ADJ
ejpam-4412	461	16	operators	operator	NOUN
ejpam-4412	461	17	.	.	PUNCT
ejpam-4412	462	1	4	4	X
ejpam-4412	462	2	.	.	X
ejpam-4412	462	3	the	the	DET
ejpam-4412	462	4	fuglede	fuglede	PROPN
ejpam-4412	462	5	-	-	PUNCT
ejpam-4412	462	6	putnam	putnam	NOUN
ejpam-4412	462	7	theorem	theorem	NOUN
ejpam-4412	462	8	we	we	PRON
ejpam-4412	462	9	offer	offer	VERB
ejpam-4412	462	10	various	various	ADJ
ejpam-4412	462	11	results	result	NOUN
ejpam-4412	462	12	related	relate	VERB
ejpam-4412	462	13	to	to	ADP
ejpam-4412	462	14	the	the	DET
ejpam-4412	462	15	fuglede	fuglede	NOUN
ejpam-4412	462	16	-	-	PUNCT
ejpam-4412	462	17	putnam	putnam	NOUN
ejpam-4412	462	18	theorem	theorem	NOUN
ejpam-4412	462	19	in	in	ADP
ejpam-4412	462	20	this	this	DET
ejpam-4412	462	21	section	section	NOUN
ejpam-4412	462	22	.	.	PUNCT
ejpam-4412	463	1	if	if	SCONJ
ejpam-4412	463	2	t	t	PROPN
ejpam-4412	463	3	∗x	∗x	NOUN
ejpam-4412	463	4	=	=	PUNCT
ejpam-4412	463	5	xs∗	xs∗	ADJ
ejpam-4412	463	6	whenever	whenever	SCONJ
ejpam-4412	463	7	tx	tx	PROPN
ejpam-4412	463	8	=	=	PUNCT
ejpam-4412	463	9	xs	xs	PROPN
ejpam-4412	463	10	for	for	ADP
ejpam-4412	463	11	every	every	DET
ejpam-4412	463	12	x	x	PROPN
ejpam-4412	463	13	∈	∈	PROPN
ejpam-4412	463	14	b(k	b(k	PROPN
ejpam-4412	463	15	,	,	PUNCT
ejpam-4412	463	16	h	h	NOUN
ejpam-4412	463	17	)	)	PUNCT
ejpam-4412	463	18	,	,	PUNCT
ejpam-4412	463	19	a	a	DET
ejpam-4412	463	20	pair	pair	NOUN
ejpam-4412	463	21	(	(	PUNCT
ejpam-4412	463	22	t	t	PROPN
ejpam-4412	463	23	,	,	PUNCT
ejpam-4412	463	24	s	s	PART
ejpam-4412	463	25	)	)	PUNCT
ejpam-4412	463	26	is	be	AUX
ejpam-4412	463	27	said	say	VERB
ejpam-4412	463	28	to	to	PART
ejpam-4412	463	29	have	have	VERB
ejpam-4412	463	30	the	the	DET
ejpam-4412	463	31	fuglede	fuglede	ADJ
ejpam-4412	463	32	-	-	PUNCT
ejpam-4412	463	33	putnam	putnam	NOUN
ejpam-4412	463	34	property	property	NOUN
ejpam-4412	463	35	.	.	PUNCT
ejpam-4412	464	1	in	in	ADP
ejpam-4412	464	2	operator	operator	NOUN
ejpam-4412	464	3	theory	theory	NOUN
ejpam-4412	464	4	,	,	PUNCT
ejpam-4412	464	5	the	the	DET
ejpam-4412	464	6	fuglede	fuglede	NOUN
ejpam-4412	464	7	-	-	PUNCT
ejpam-4412	464	8	putnam	putnam	NOUN
ejpam-4412	464	9	theorem	theorem	NOUN
ejpam-4412	464	10	is	be	AUX
ejpam-4412	464	11	wellknown	wellknown	ADJ
ejpam-4412	464	12	.	.	PUNCT
ejpam-4412	465	1	it	it	PRON
ejpam-4412	465	2	claims	claim	VERB
ejpam-4412	465	3	that	that	SCONJ
ejpam-4412	465	4	the	the	DET
ejpam-4412	465	5	pair	pair	NOUN
ejpam-4412	465	6	(	(	PUNCT
ejpam-4412	465	7	t	t	PROPN
ejpam-4412	465	8	,	,	PUNCT
ejpam-4412	465	9	s	s	PART
ejpam-4412	465	10	)	)	PUNCT
ejpam-4412	465	11	possesses	possess	VERB
ejpam-4412	465	12	the	the	DET
ejpam-4412	465	13	fuglede	fuglede	NOUN
ejpam-4412	465	14	-	-	PUNCT
ejpam-4412	465	15	putnam	putnam	NOUN
ejpam-4412	465	16	property	property	NOUN
ejpam-4412	465	17	for	for	ADP
ejpam-4412	465	18	any	any	DET
ejpam-4412	465	19	normal	normal	ADJ
ejpam-4412	465	20	operators	operator	NOUN
ejpam-4412	465	21	t	t	PROPN
ejpam-4412	465	22	and	and	CCONJ
ejpam-4412	465	23	s.	s.	PROPN
ejpam-4412	465	24	there	there	PRON
ejpam-4412	465	25	are	be	VERB
ejpam-4412	465	26	several	several	ADJ
ejpam-4412	465	27	generalizations	generalization	NOUN
ejpam-4412	465	28	of	of	ADP
ejpam-4412	465	29	this	this	DET
ejpam-4412	465	30	theorem	theorem	NOUN
ejpam-4412	465	31	,	,	PUNCT
ejpam-4412	465	32	the	the	DET
ejpam-4412	465	33	majority	majority	NOUN
ejpam-4412	465	34	of	of	ADP
ejpam-4412	465	35	which	which	PRON
ejpam-4412	465	36	loosen	loosen	VERB
ejpam-4412	465	37	the	the	DET
ejpam-4412	465	38	normality	normality	NOUN
ejpam-4412	465	39	of	of	ADP
ejpam-4412	465	40	t	t	PROPN
ejpam-4412	465	41	and	and	CCONJ
ejpam-4412	465	42	s	s	PART
ejpam-4412	465	43	;	;	PUNCT
ejpam-4412	465	44	see	see	VERB
ejpam-4412	465	45	,	,	PUNCT
ejpam-4412	465	46	for	for	ADP
ejpam-4412	465	47	example	example	NOUN
ejpam-4412	465	48	,	,	PUNCT
ejpam-4412	465	49	[	[	X
ejpam-4412	465	50	22–24	22–24	NUM
ejpam-4412	465	51	,	,	PUNCT
ejpam-4412	465	52	27	27	NUM
ejpam-4412	465	53	,	,	PUNCT
ejpam-4412	465	54	28	28	NUM
ejpam-4412	465	55	]	]	PUNCT
ejpam-4412	465	56	,	,	PUNCT
ejpam-4412	465	57	and	and	CCONJ
ejpam-4412	465	58	some	some	DET
ejpam-4412	465	59	references	reference	NOUN
ejpam-4412	465	60	therein	therein	ADV
ejpam-4412	465	61	and	and	CCONJ
ejpam-4412	465	62	for	for	ADP
ejpam-4412	465	63	more	more	ADJ
ejpam-4412	465	64	details	detail	NOUN
ejpam-4412	465	65	(	(	PUNCT
ejpam-4412	465	66	see	see	VERB
ejpam-4412	465	67	[	[	X
ejpam-4412	465	68	3],[5],[4	3],[5],[4	NUM
ejpam-4412	465	69	]	]	X
ejpam-4412	465	70	)	)	PUNCT
ejpam-4412	465	71	.	.	PUNCT
ejpam-4412	466	1	the	the	DET
ejpam-4412	466	2	fuglede	fuglede	PROPN
ejpam-4412	466	3	-	-	PUNCT
ejpam-4412	466	4	putnam	putnam	NOUN
ejpam-4412	466	5	theorem	theorem	NOUN
ejpam-4412	466	6	is	be	AUX
ejpam-4412	466	7	the	the	DET
ejpam-4412	466	8	subject	subject	NOUN
ejpam-4412	466	9	of	of	ADP
ejpam-4412	466	10	the	the	DET
ejpam-4412	466	11	next	next	ADJ
ejpam-4412	466	12	lemma	lemma	PROPN
ejpam-4412	466	13	,	,	PUNCT
ejpam-4412	466	14	which	which	PRON
ejpam-4412	466	15	we	we	PRON
ejpam-4412	466	16	will	will	AUX
ejpam-4412	466	17	require	require	VERB
ejpam-4412	466	18	in	in	ADP
ejpam-4412	466	19	the	the	DET
ejpam-4412	466	20	future	future	NOUN
ejpam-4412	466	21	.	.	PUNCT
ejpam-4412	467	1	lemma	lemma	PROPN
ejpam-4412	467	2	14	14	NUM
ejpam-4412	467	3	.	.	PUNCT
ejpam-4412	468	1	(	(	PUNCT
ejpam-4412	468	2	[	[	X
ejpam-4412	468	3	29	29	NUM
ejpam-4412	468	4	]	]	PUNCT
ejpam-4412	468	5	)	)	PUNCT
ejpam-4412	468	6	let	let	VERB
ejpam-4412	468	7	t	t	PROPN
ejpam-4412	468	8	∈	∈	PROPN
ejpam-4412	468	9	b(h	b(h	PROPN
ejpam-4412	468	10	)	)	PUNCT
ejpam-4412	468	11	and	and	CCONJ
ejpam-4412	468	12	s	s	PROPN
ejpam-4412	468	13	∈	∈	PROPN
ejpam-4412	468	14	b(k	b(k	PROPN
ejpam-4412	468	15	)	)	PUNCT
ejpam-4412	468	16	.	.	PUNCT
ejpam-4412	469	1	then	then	ADV
ejpam-4412	469	2	the	the	DET
ejpam-4412	469	3	following	follow	VERB
ejpam-4412	469	4	assertions	assertion	NOUN
ejpam-4412	469	5	equivalent	equivalent	ADJ
ejpam-4412	469	6	.	.	PUNCT
ejpam-4412	470	1	(	(	PUNCT
ejpam-4412	470	2	i	i	NOUN
ejpam-4412	470	3	)	)	PUNCT
ejpam-4412	470	4	the	the	DET
ejpam-4412	470	5	pair	pair	NOUN
ejpam-4412	470	6	(	(	PUNCT
ejpam-4412	470	7	t	t	PROPN
ejpam-4412	470	8	,	,	PUNCT
ejpam-4412	470	9	s	s	PART
ejpam-4412	470	10	)	)	PUNCT
ejpam-4412	470	11	has	have	VERB
ejpam-4412	470	12	the	the	DET
ejpam-4412	470	13	fuglede	fuglede	ADJ
ejpam-4412	470	14	-	-	PUNCT
ejpam-4412	470	15	putnam	putnam	NOUN
ejpam-4412	470	16	property	property	NOUN
ejpam-4412	470	17	.	.	PUNCT
ejpam-4412	471	1	(	(	PUNCT
ejpam-4412	471	2	ii	ii	NOUN
ejpam-4412	471	3	)	)	PUNCT
ejpam-4412	471	4	if	if	SCONJ
ejpam-4412	471	5	tx	tx	PROPN
ejpam-4412	471	6	=	=	SYM
ejpam-4412	471	7	xs	xs	PROPN
ejpam-4412	471	8	,	,	PUNCT
ejpam-4412	471	9	then	then	ADV
ejpam-4412	471	10	ran(x	ran(x	PROPN
ejpam-4412	471	11	)	)	PUNCT
ejpam-4412	471	12	reduces	reduce	VERB
ejpam-4412	471	13	t	t	NOUN
ejpam-4412	471	14	,	,	PUNCT
ejpam-4412	471	15	ker(x)⊥	ker(x)⊥	PROPN
ejpam-4412	471	16	reduces	reduce	VERB
ejpam-4412	471	17	s	s	PRON
ejpam-4412	471	18	,	,	PUNCT
ejpam-4412	471	19	and	and	CCONJ
ejpam-4412	471	20	t	t	NOUN
ejpam-4412	471	21	|	|	ADV
ejpam-4412	471	22	ran(x	ran(x	PROPN
ejpam-4412	471	23	)	)	PUNCT
ejpam-4412	471	24	,	,	PUNCT
ejpam-4412	471	25	s|ker(x)⊥	s|ker(x)⊥	NOUN
ejpam-4412	471	26	are	be	AUX
ejpam-4412	471	27	unitarily	unitarily	ADV
ejpam-4412	471	28	equivalent	equivalent	ADJ
ejpam-4412	471	29	normal	normal	ADJ
ejpam-4412	471	30	operators	operator	NOUN
ejpam-4412	471	31	.	.	PUNCT
ejpam-4412	472	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	472	2	,	,	PUNCT
ejpam-4412	472	3	n.	n.	NOUN
ejpam-4412	472	4	h.	h.	PROPN
ejpam-4412	472	5	altaweel	altaweel	PROPN
ejpam-4412	472	6	/	/	SYM
ejpam-4412	472	7	eur	eur	PROPN
ejpam-4412	472	8	.	.	PUNCT
ejpam-4412	473	1	j.	j.	PROPN
ejpam-4412	473	2	pure	pure	PROPN
ejpam-4412	473	3	appl	appl	PROPN
ejpam-4412	473	4	.	.	PROPN
ejpam-4412	473	5	math	math	PROPN
ejpam-4412	473	6	,	,	PUNCT
ejpam-4412	473	7	15	15	NUM
ejpam-4412	473	8	(	(	PUNCT
ejpam-4412	473	9	3	3	NUM
ejpam-4412	473	10	)	)	PUNCT
ejpam-4412	473	11	(	(	PUNCT
ejpam-4412	473	12	2022	2022	NUM
ejpam-4412	473	13	)	)	PUNCT
ejpam-4412	473	14	,	,	PUNCT
ejpam-4412	473	15	1067	1067	NUM
ejpam-4412	473	16	-	-	SYM
ejpam-4412	473	17	1089	1089	NUM
ejpam-4412	473	18	1082	1082	NUM
ejpam-4412	473	19	remark	remark	NOUN
ejpam-4412	473	20	1	1	NUM
ejpam-4412	473	21	.	.	PUNCT
ejpam-4412	474	1	a	a	DET
ejpam-4412	474	2	necessary	necessary	ADJ
ejpam-4412	474	3	condition	condition	NOUN
ejpam-4412	474	4	for	for	ADP
ejpam-4412	474	5	the	the	DET
ejpam-4412	474	6	pair	pair	NOUN
ejpam-4412	474	7	(	(	PUNCT
ejpam-4412	474	8	t	t	PROPN
ejpam-4412	474	9	,	,	PUNCT
ejpam-4412	474	10	t	t	PROPN
ejpam-4412	474	11	∗	∗	NOUN
ejpam-4412	474	12	)	)	PUNCT
ejpam-4412	474	13	to	to	PART
ejpam-4412	474	14	satisfy	satisfy	VERB
ejpam-4412	474	15	fuglede	fuglede	PROPN
ejpam-4412	474	16	-	-	PUNCT
ejpam-4412	474	17	putnam	putnam	NOUN
ejpam-4412	474	18	’s	’s	PART
ejpam-4412	474	19	theorem	theorem	NOUN
ejpam-4412	474	20	is	be	AUX
ejpam-4412	474	21	ker(t	ker(t	NOUN
ejpam-4412	474	22	)	)	PUNCT
ejpam-4412	475	1	⊂	⊂	PROPN
ejpam-4412	475	2	ker(t	ker(t	NOUN
ejpam-4412	475	3	∗	∗	NOUN
ejpam-4412	475	4	)	)	PUNCT
ejpam-4412	475	5	.	.	PUNCT
ejpam-4412	476	1	since	since	SCONJ
ejpam-4412	476	2	for	for	ADP
ejpam-4412	476	3	a	a	DET
ejpam-4412	476	4	class	class	NOUN
ejpam-4412	476	5	p	p	NOUN
ejpam-4412	476	6	-	-	PUNCT
ejpam-4412	476	7	wa(s	wa(s	NUM
ejpam-4412	476	8	,	,	PUNCT
ejpam-4412	476	9	t	t	NOUN
ejpam-4412	476	10	)	)	PUNCT
ejpam-4412	476	11	operator	operator	NOUN
ejpam-4412	476	12	this	this	PRON
ejpam-4412	476	13	is	be	AUX
ejpam-4412	476	14	not	not	PART
ejpam-4412	476	15	always	always	ADV
ejpam-4412	476	16	true	true	ADJ
ejpam-4412	476	17	,	,	PUNCT
ejpam-4412	476	18	class	class	NOUN
ejpam-4412	476	19	p	p	NOUN
ejpam-4412	476	20	-	-	PUNCT
ejpam-4412	476	21	wa(s	wa(s	NUM
ejpam-4412	476	22	,	,	PUNCT
ejpam-4412	476	23	t	t	NOUN
ejpam-4412	476	24	)	)	PUNCT
ejpam-4412	476	25	operator	operator	NOUN
ejpam-4412	476	26	do	do	AUX
ejpam-4412	476	27	not	not	PART
ejpam-4412	476	28	fuglede	fuglede	PROPN
ejpam-4412	476	29	-	-	PUNCT
ejpam-4412	476	30	putnam	putnam	PROPN
ejpam-4412	476	31	’s	’s	PART
ejpam-4412	476	32	theorem	theorem	NOUN
ejpam-4412	476	33	.	.	PUNCT
ejpam-4412	477	1	for	for	ADP
ejpam-4412	477	2	example	example	NOUN
ejpam-4412	477	3	,	,	PUNCT
ejpam-4412	477	4	if	if	SCONJ
ejpam-4412	477	5	p	p	NOUN
ejpam-4412	477	6	is	be	AUX
ejpam-4412	477	7	the	the	DET
ejpam-4412	477	8	orthogonal	orthogonal	ADJ
ejpam-4412	477	9	projection	projection	NOUN
ejpam-4412	477	10	onto	onto	ADP
ejpam-4412	477	11	ker(t	ker(t	NOUN
ejpam-4412	477	12	)	)	PUNCT
ejpam-4412	477	13	,	,	PUNCT
ejpam-4412	477	14	with	with	ADP
ejpam-4412	477	15	t	t	PROPN
ejpam-4412	477	16	is	be	AUX
ejpam-4412	477	17	class	class	NOUN
ejpam-4412	477	18	p	p	NOUN
ejpam-4412	477	19	-	-	PUNCT
ejpam-4412	477	20	wa(s	wa(s	NUM
ejpam-4412	477	21	,	,	PUNCT
ejpam-4412	477	22	t	t	PROPN
ejpam-4412	477	23	)	)	PUNCT
ejpam-4412	477	24	,	,	PUNCT
ejpam-4412	477	25	then	then	ADV
ejpam-4412	477	26	tp	tp	ADP
ejpam-4412	477	27	=	=	SYM
ejpam-4412	477	28	pt	pt	X
ejpam-4412	477	29	∗	∗	NOUN
ejpam-4412	477	30	but	but	CCONJ
ejpam-4412	477	31	t	t	PROPN
ejpam-4412	477	32	∗p	∗p	PROPN
ejpam-4412	477	33	̸=	̸=	PROPN
ejpam-4412	477	34	pt	pt	PROPN
ejpam-4412	477	35	.	.	PUNCT
ejpam-4412	478	1	the	the	DET
ejpam-4412	478	2	following	following	ADJ
ejpam-4412	478	3	result	result	NOUN
ejpam-4412	478	4	(	(	PUNCT
ejpam-4412	478	5	corollary	corollary	ADJ
ejpam-4412	478	6	6	6	NUM
ejpam-4412	478	7	)	)	PUNCT
ejpam-4412	478	8	prove	prove	VERB
ejpam-4412	478	9	that	that	SCONJ
ejpam-4412	478	10	if	if	SCONJ
ejpam-4412	478	11	t	t	PROPN
ejpam-4412	478	12	∗	∗	NOUN
ejpam-4412	478	13	,	,	PUNCT
ejpam-4412	478	14	s	s	VERB
ejpam-4412	478	15	are	be	AUX
ejpam-4412	478	16	p	p	ADJ
ejpam-4412	478	17	-	-	PUNCT
ejpam-4412	478	18	class	class	NOUN
ejpam-4412	478	19	a(s	a(s	PROPN
ejpam-4412	478	20	,	,	PUNCT
ejpam-4412	478	21	t	t	PROPN
ejpam-4412	478	22	)	)	PUNCT
ejpam-4412	478	23	operators	operator	NOUN
ejpam-4412	478	24	for	for	ADP
ejpam-4412	478	25	0	0	NUM
ejpam-4412	478	26	<	<	X
ejpam-4412	478	27	s	s	PROPN
ejpam-4412	478	28	,	,	PUNCT
ejpam-4412	478	29	t	t	PROPN
ejpam-4412	478	30	,	,	PUNCT
ejpam-4412	478	31	s+	s+	ADP
ejpam-4412	478	32	t	t	PROPN
ejpam-4412	478	33	=	=	SYM
ejpam-4412	478	34	1	1	NUM
ejpam-4412	478	35	and	and	CCONJ
ejpam-4412	478	36	0	0	NUM
ejpam-4412	478	37	<	<	X
ejpam-4412	478	38	p	p	X
ejpam-4412	478	39	≤	≤	NUM
ejpam-4412	478	40	1	1	NUM
ejpam-4412	478	41	such	such	ADJ
ejpam-4412	478	42	that	that	DET
ejpam-4412	478	43	ker(t	ker(t	NOUN
ejpam-4412	478	44	∗	∗	NOUN
ejpam-4412	478	45	)	)	PUNCT
ejpam-4412	478	46	reduces	reduce	VERB
ejpam-4412	478	47	t	t	NOUN
ejpam-4412	478	48	∗	∗	NOUN
ejpam-4412	478	49	and	and	CCONJ
ejpam-4412	478	50	ker(s	ker(s	NOUN
ejpam-4412	478	51	)	)	PUNCT
ejpam-4412	478	52	reduces	reduce	VERB
ejpam-4412	478	53	s	s	PROPN
ejpam-4412	478	54	,	,	PUNCT
ejpam-4412	478	55	then	then	ADV
ejpam-4412	478	56	the	the	DET
ejpam-4412	478	57	pair	pair	NOUN
ejpam-4412	478	58	(	(	PUNCT
ejpam-4412	478	59	t	t	PROPN
ejpam-4412	478	60	,	,	PUNCT
ejpam-4412	478	61	s	s	PART
ejpam-4412	478	62	)	)	PUNCT
ejpam-4412	478	63	satisfy	satisfy	VERB
ejpam-4412	478	64	fuglede	fuglede	PROPN
ejpam-4412	478	65	-	-	PUNCT
ejpam-4412	478	66	putnam	putnam	PROPN
ejpam-4412	478	67	’s	’s	PART
ejpam-4412	478	68	theorem	theorem	PROPN
ejpam-4412	478	69	.	.	PUNCT
ejpam-4412	478	70	theorem	theorem	PROPN
ejpam-4412	478	71	10	10	NUM
ejpam-4412	478	72	.	.	PUNCT
ejpam-4412	479	1	let	let	AUX
ejpam-4412	479	2	t	t	PROPN
ejpam-4412	479	3	∈	∈	PROPN
ejpam-4412	479	4	b(h	b(h	PROPN
ejpam-4412	479	5	)	)	PUNCT
ejpam-4412	479	6	be	be	AUX
ejpam-4412	479	7	class	class	NOUN
ejpam-4412	479	8	p	p	NOUN
ejpam-4412	479	9	-	-	PUNCT
ejpam-4412	479	10	wa(s	wa(s	NUM
ejpam-4412	479	11	,	,	PUNCT
ejpam-4412	479	12	t	t	NOUN
ejpam-4412	479	13	)	)	PUNCT
ejpam-4412	479	14	operator	operator	NOUN
ejpam-4412	479	15	for	for	ADP
ejpam-4412	479	16	0	0	NUM
ejpam-4412	479	17	<	<	X
ejpam-4412	479	18	s	s	PROPN
ejpam-4412	479	19	,	,	PUNCT
ejpam-4412	479	20	t	t	PROPN
ejpam-4412	479	21	,	,	PUNCT
ejpam-4412	479	22	s	s	PART
ejpam-4412	479	23	+	+	NUM
ejpam-4412	479	24	t	t	X
ejpam-4412	479	25	=	=	SYM
ejpam-4412	479	26	1	1	NUM
ejpam-4412	479	27	and	and	CCONJ
ejpam-4412	479	28	0	0	NUM
ejpam-4412	480	1	<	<	X
ejpam-4412	480	2	p	p	X
ejpam-4412	480	3	≤	≤	NUM
ejpam-4412	480	4	1	1	NUM
ejpam-4412	480	5	and	and	CCONJ
ejpam-4412	480	6	ker(t	ker(t	NOUN
ejpam-4412	480	7	)	)	PUNCT
ejpam-4412	480	8	⊂	⊂	PROPN
ejpam-4412	480	9	ker(t	ker(t	NOUN
ejpam-4412	480	10	∗	∗	NOUN
ejpam-4412	480	11	)	)	PUNCT
ejpam-4412	480	12	.	.	PUNCT
ejpam-4412	481	1	if	if	SCONJ
ejpam-4412	481	2	l	l	NOUN
ejpam-4412	481	3	is	be	AUX
ejpam-4412	481	4	self	self	NOUN
ejpam-4412	481	5	-	-	PUNCT
ejpam-4412	481	6	adjoint	adjoint	NOUN
ejpam-4412	481	7	and	and	CCONJ
ejpam-4412	481	8	tl	tl	PROPN
ejpam-4412	482	1	=	=	PUNCT
ejpam-4412	482	2	lt	lt	DET
ejpam-4412	482	3	∗	∗	NOUN
ejpam-4412	482	4	,	,	PUNCT
ejpam-4412	482	5	then	then	ADV
ejpam-4412	482	6	t	t	PROPN
ejpam-4412	482	7	∗l	∗l	NOUN
ejpam-4412	482	8	=	=	SYM
ejpam-4412	482	9	lt	lt	PROPN
ejpam-4412	482	10	.	.	PROPN
ejpam-4412	482	11	proof	proof	NOUN
ejpam-4412	482	12	.	.	PUNCT
ejpam-4412	483	1	since	since	SCONJ
ejpam-4412	483	2	ker(t	ker(t	NOUN
ejpam-4412	483	3	)	)	PUNCT
ejpam-4412	484	1	⊂	⊂	PROPN
ejpam-4412	484	2	ker(t	ker(t	NOUN
ejpam-4412	484	3	∗	∗	NOUN
ejpam-4412	484	4	)	)	PUNCT
ejpam-4412	484	5	and	and	CCONJ
ejpam-4412	484	6	tl	tl	PROPN
ejpam-4412	484	7	=	=	PUNCT
ejpam-4412	485	1	lt	lt	DET
ejpam-4412	485	2	∗	∗	NOUN
ejpam-4412	485	3	,	,	PUNCT
ejpam-4412	485	4	ker(t	ker(t	NOUN
ejpam-4412	485	5	)	)	PUNCT
ejpam-4412	485	6	reduces	reduce	VERB
ejpam-4412	485	7	t	t	PROPN
ejpam-4412	485	8	and	and	CCONJ
ejpam-4412	485	9	l.	l.	PROPN
ejpam-4412	485	10	hence	hence	ADV
ejpam-4412	485	11	t	t	PROPN
ejpam-4412	486	1	=	=	PUNCT
ejpam-4412	486	2	t1	t1	PROPN
ejpam-4412	486	3	⊕	⊕	PROPN
ejpam-4412	486	4	0	0	NUM
ejpam-4412	486	5	,	,	PUNCT
ejpam-4412	486	6	l	l	PROPN
ejpam-4412	486	7	=	=	PROPN
ejpam-4412	486	8	l1	l1	PROPN
ejpam-4412	486	9	⊕	⊕	PROPN
ejpam-4412	486	10	l2	l2	NOUN
ejpam-4412	486	11	on	on	ADP
ejpam-4412	486	12	h	h	NOUN
ejpam-4412	486	13	=	=	NOUN
ejpam-4412	486	14	ran(t	ran(t	PROPN
ejpam-4412	486	15	∗)⊕	∗)⊕	ADP
ejpam-4412	486	16	ker(t	ker(t	PROPN
ejpam-4412	486	17	)	)	PUNCT
ejpam-4412	486	18	,	,	PUNCT
ejpam-4412	486	19	t1l1	t1l1	X
ejpam-4412	486	20	=	=	SYM
ejpam-4412	486	21	l1	l1	PROPN
ejpam-4412	486	22	t	t	PROPN
ejpam-4412	486	23	∗	∗	NOUN
ejpam-4412	486	24	and	and	CCONJ
ejpam-4412	486	25	{	{	PUNCT
ejpam-4412	486	26	0	0	NUM
ejpam-4412	486	27	}	}	PUNCT
ejpam-4412	486	28	=	=	SYM
ejpam-4412	486	29	ker(t1	ker(t1	PROPN
ejpam-4412	486	30	)	)	PUNCT
ejpam-4412	487	1	⊂	⊂	PROPN
ejpam-4412	487	2	ker(t	ker(t	PROPN
ejpam-4412	487	3	∗	∗	NOUN
ejpam-4412	487	4	1	1	NUM
ejpam-4412	487	5	)	)	PUNCT
ejpam-4412	487	6	.	.	PUNCT
ejpam-4412	488	1	since	since	SCONJ
ejpam-4412	488	2	ran(l1	ran(l1	PROPN
ejpam-4412	488	3	)	)	PUNCT
ejpam-4412	488	4	is	be	AUX
ejpam-4412	488	5	invariant	invariant	ADJ
ejpam-4412	488	6	under	under	ADP
ejpam-4412	488	7	t1	t1	NOUN
ejpam-4412	488	8	and	and	CCONJ
ejpam-4412	488	9	reduces	reduce	VERB
ejpam-4412	488	10	l1	l1	PROPN
ejpam-4412	488	11	,	,	PUNCT
ejpam-4412	488	12	t	t	NOUN
ejpam-4412	488	13	=	=	PUNCT
ejpam-4412	488	14	(	(	PUNCT
ejpam-4412	488	15	t11	t11	PROPN
ejpam-4412	488	16	s	s	PROPN
ejpam-4412	488	17	0	0	PROPN
ejpam-4412	488	18	t22	t22	PROPN
ejpam-4412	488	19	)	)	PUNCT
ejpam-4412	488	20	,	,	PUNCT
ejpam-4412	488	21	l1	l1	PROPN
ejpam-4412	488	22	=	=	SYM
ejpam-4412	488	23	l11	l11	PROPN
ejpam-4412	488	24	⊕	⊕	PROPN
ejpam-4412	488	25	0	0	NUM
ejpam-4412	489	1	on	on	ADP
ejpam-4412	489	2	h	h	NOUN
ejpam-4412	489	3	=	=	NOUN
ejpam-4412	489	4	ran(t	ran(t	NOUN
ejpam-4412	489	5	∗	∗	NOUN
ejpam-4412	489	6	)	)	PUNCT
ejpam-4412	489	7	=	=	SYM
ejpam-4412	489	8	ran(l1)⊕	ran(l1)⊕	NOUN
ejpam-4412	489	9	ker(l1	ker(l1	PROPN
ejpam-4412	489	10	)	)	PUNCT
ejpam-4412	489	11	.	.	PUNCT
ejpam-4412	490	1	t11	t11	NOUN
ejpam-4412	490	2	is	be	AUX
ejpam-4412	490	3	an	an	DET
ejpam-4412	490	4	injective	injective	ADJ
ejpam-4412	490	5	class	class	NOUN
ejpam-4412	490	6	p	p	NOUN
ejpam-4412	490	7	-	-	PUNCT
ejpam-4412	490	8	wa(s	wa(s	NUM
ejpam-4412	490	9	,	,	PUNCT
ejpam-4412	490	10	t	t	NOUN
ejpam-4412	490	11	)	)	PUNCT
ejpam-4412	490	12	operator	operator	NOUN
ejpam-4412	490	13	by	by	ADP
ejpam-4412	490	14	lemma	lemma	PROPN
ejpam-4412	490	15	6	6	NUM
ejpam-4412	490	16	and	and	CCONJ
ejpam-4412	490	17	l11	l11	PROPN
ejpam-4412	490	18	is	be	AUX
ejpam-4412	490	19	an	an	DET
ejpam-4412	490	20	injective	injective	ADJ
ejpam-4412	490	21	self	self	NOUN
ejpam-4412	490	22	-	-	PUNCT
ejpam-4412	490	23	adjoint	adjoint	NOUN
ejpam-4412	490	24	operator	operator	NOUN
ejpam-4412	490	25	(	(	PUNCT
ejpam-4412	490	26	hence	hence	ADV
ejpam-4412	490	27	it	it	PRON
ejpam-4412	490	28	has	have	AUX
ejpam-4412	490	29	dense	dense	ADJ
ejpam-4412	490	30	range	range	NOUN
ejpam-4412	490	31	)	)	PUNCT
ejpam-4412	490	32	such	such	ADJ
ejpam-4412	490	33	that	that	DET
ejpam-4412	490	34	t11l11	t11l11	PROPN
ejpam-4412	490	35	=	=	SYM
ejpam-4412	490	36	l11	l11	PROPN
ejpam-4412	490	37	t	t	PROPN
ejpam-4412	490	38	∗	∗	PROPN
ejpam-4412	490	39	11	11	NUM
ejpam-4412	490	40	.	.	PUNCT
ejpam-4412	491	1	let	let	VERB
ejpam-4412	491	2	t11	t11	NOUN
ejpam-4412	491	3	=	=	PUNCT
ejpam-4412	491	4	v11|t11|	v11|t11|	NOUN
ejpam-4412	491	5	be	be	AUX
ejpam-4412	491	6	the	the	DET
ejpam-4412	491	7	polar	polar	ADJ
ejpam-4412	491	8	decomposition	decomposition	NOUN
ejpam-4412	491	9	of	of	ADP
ejpam-4412	491	10	t11	t11	NOUN
ejpam-4412	491	11	and	and	CCONJ
ejpam-4412	491	12	t11(s	t11(s	PROPN
ejpam-4412	491	13	,	,	PUNCT
ejpam-4412	491	14	t	t	PROPN
ejpam-4412	491	15	)	)	PUNCT
ejpam-4412	491	16	=	=	SYM
ejpam-4412	492	1	|t11|sv11|t11|t	|t11|sv11|t11|t	PROPN
ejpam-4412	492	2	,	,	PUNCT
ejpam-4412	492	3	w	w	NOUN
ejpam-4412	492	4	=	=	PUNCT
ejpam-4412	492	5	|t11|sl11|t11|s	|t11|sl11|t11|s	NOUN
ejpam-4412	492	6	.	.	PUNCT
ejpam-4412	493	1	then	then	ADV
ejpam-4412	493	2	t11(s	t11(	NOUN
ejpam-4412	493	3	,	,	PUNCT
ejpam-4412	493	4	t)w	t)w	PUNCT
ejpam-4412	493	5	=	=	SYM
ejpam-4412	493	6	|t11|sv11|t11|t|t11|sl11|t11|s	|t11|sv11|t11|t|t11|sl11|t11|s	X
ejpam-4412	493	7	=	=	SYM
ejpam-4412	493	8	|t11|st11l11|t11|s	|t11|st11l11|t11|s	PROPN
ejpam-4412	493	9	=	=	PUNCT
ejpam-4412	493	10	|t11|sl11	|t11|sl11	NOUN
ejpam-4412	493	11	t	t	PROPN
ejpam-4412	493	12	∗	∗	X
ejpam-4412	493	13	11|t11|s	11|t11|s	NUM
ejpam-4412	493	14	=	=	SYM
ejpam-4412	493	15	|t11|sl11|t11|s|t11|tv	|t11|sl11|t11|s|t11|tv	NOUN
ejpam-4412	493	16	∗	∗	X
ejpam-4412	493	17	11|t11|s	11|t11|s	NUM
ejpam-4412	493	18	=	=	SYM
ejpam-4412	493	19	w	w	PROPN
ejpam-4412	493	20	(	(	PUNCT
ejpam-4412	493	21	t11(s	t11(s	PROPN
ejpam-4412	493	22	,	,	PUNCT
ejpam-4412	493	23	t	t	PROPN
ejpam-4412	493	24	)	)	PUNCT
ejpam-4412	493	25	)	)	PUNCT
ejpam-4412	494	1	∗.	∗.	PUNCT
ejpam-4412	494	2	since	since	SCONJ
ejpam-4412	494	3	t11(s	t11(	NOUN
ejpam-4412	494	4	,	,	PUNCT
ejpam-4412	494	5	t	t	PROPN
ejpam-4412	494	6	)	)	PUNCT
ejpam-4412	494	7	is	be	AUX
ejpam-4412	494	8	min{sp	min{sp	NUM
ejpam-4412	494	9	,	,	PUNCT
ejpam-4412	494	10	tp}-hyponormal	tp}-hyponormal	ADJ
ejpam-4412	494	11	and	and	CCONJ
ejpam-4412	494	12	ran(w	ran(w	PROPN
ejpam-4412	494	13	)	)	PUNCT
ejpam-4412	494	14	is	be	AUX
ejpam-4412	494	15	dense	dense	ADJ
ejpam-4412	494	16	(	(	PUNCT
ejpam-4412	494	17	because	because	SCONJ
ejpam-4412	494	18	ker(w	ker(w	PROPN
ejpam-4412	494	19	)	)	PUNCT
ejpam-4412	494	20	=	=	PUNCT
ejpam-4412	494	21	{	{	PUNCT
ejpam-4412	494	22	0	0	NUM
ejpam-4412	494	23	}	}	PUNCT
ejpam-4412	494	24	)	)	PUNCT
ejpam-4412	494	25	,	,	PUNCT
ejpam-4412	494	26	t11(s	t11(s	PROPN
ejpam-4412	494	27	,	,	PUNCT
ejpam-4412	494	28	t	t	PROPN
ejpam-4412	494	29	)	)	PUNCT
ejpam-4412	494	30	is	be	AUX
ejpam-4412	494	31	normal	normal	ADJ
ejpam-4412	494	32	by	by	ADP
ejpam-4412	494	33	[	[	X
ejpam-4412	494	34	12	12	NUM
ejpam-4412	494	35	,	,	PUNCT
ejpam-4412	494	36	theorem	theorem	VERB
ejpam-4412	494	37	7	7	NUM
ejpam-4412	494	38	]	]	PUNCT
ejpam-4412	494	39	.	.	PUNCT
ejpam-4412	495	1	hence	hence	ADV
ejpam-4412	495	2	t11	t11	PROPN
ejpam-4412	495	3	is	be	AUX
ejpam-4412	495	4	normal	normal	ADJ
ejpam-4412	495	5	and	and	CCONJ
ejpam-4412	495	6	t11	t11	NOUN
ejpam-4412	495	7	=	=	PUNCT
ejpam-4412	495	8	t11(s	t11(s	PROPN
ejpam-4412	495	9	,	,	PUNCT
ejpam-4412	495	10	t	t	PROPN
ejpam-4412	495	11	)	)	PUNCT
ejpam-4412	495	12	by	by	ADP
ejpam-4412	495	13	corollary	corollary	ADJ
ejpam-4412	495	14	1	1	NUM
ejpam-4412	495	15	.	.	PUNCT
ejpam-4412	496	1	then	then	ADV
ejpam-4412	496	2	ran(l1	ran(l1	PROPN
ejpam-4412	496	3	)	)	PUNCT
ejpam-4412	496	4	reduces	reduce	VERB
ejpam-4412	496	5	t1	t1	NOUN
ejpam-4412	496	6	by	by	ADP
ejpam-4412	496	7	lemma	lemma	PROPN
ejpam-4412	496	8	7	7	NUM
ejpam-4412	496	9	and	and	CCONJ
ejpam-4412	496	10	t	t	PROPN
ejpam-4412	496	11	∗	∗	NOUN
ejpam-4412	496	12	11l11	11l11	NUM
ejpam-4412	496	13	=	=	SYM
ejpam-4412	496	14	l11t11	l11t11	NOUN
ejpam-4412	496	15	by	by	ADP
ejpam-4412	496	16	lemma	lemma	PROPN
ejpam-4412	496	17	14	14	NUM
ejpam-4412	496	18	.	.	PUNCT
ejpam-4412	497	1	hence	hence	ADV
ejpam-4412	497	2	t	t	NOUN
ejpam-4412	497	3	=	=	SYM
ejpam-4412	497	4	t11	t11	PROPN
ejpam-4412	497	5	⊕	⊕	PROPN
ejpam-4412	497	6	t22	t22	PROPN
ejpam-4412	497	7	⊕	⊕	PROPN
ejpam-4412	497	8	0	0	NUM
ejpam-4412	497	9	,	,	PUNCT
ejpam-4412	497	10	l	l	NOUN
ejpam-4412	497	11	=	=	SYM
ejpam-4412	497	12	l11	l11	PROPN
ejpam-4412	497	13	⊕	⊕	PROPN
ejpam-4412	497	14	0⊕	0⊕	NUM
ejpam-4412	497	15	l2	l2	VERB
ejpam-4412	497	16	and	and	CCONJ
ejpam-4412	497	17	t	t	NOUN
ejpam-4412	497	18	∗l	∗l	NOUN
ejpam-4412	497	19	=	=	SYM
ejpam-4412	497	20	t	t	PROPN
ejpam-4412	497	21	∗	∗	NOUN
ejpam-4412	497	22	11l11	11l11	NUM
ejpam-4412	497	23	⊕	⊕	NOUN
ejpam-4412	497	24	0⊕	0⊕	NUM
ejpam-4412	497	25	0	0	NUM
ejpam-4412	497	26	=	=	SYM
ejpam-4412	497	27	l11t11	l11t11	PROPN
ejpam-4412	497	28	⊕	⊕	PROPN
ejpam-4412	497	29	0⊕	0⊕	NUM
ejpam-4412	497	30	0	0	NUM
ejpam-4412	498	1	=	=	SYM
ejpam-4412	498	2	lt	lt	PROPN
ejpam-4412	498	3	.	.	PROPN
ejpam-4412	498	4	example	example	NOUN
ejpam-4412	498	5	2	2	NUM
ejpam-4412	498	6	.	.	PUNCT
ejpam-4412	499	1	let	let	VERB
ejpam-4412	499	2	h	h	NOUN
ejpam-4412	499	3	=	=	PUNCT
ejpam-4412	500	1	∞⊕	∞⊕	PROPN
ejpam-4412	501	1	n=0	n=0	NUM
ejpam-4412	501	2	c2	c2	PROPN
ejpam-4412	501	3	and	and	CCONJ
ejpam-4412	501	4	define	define	VERB
ejpam-4412	501	5	an	an	DET
ejpam-4412	501	6	operator	operator	NOUN
ejpam-4412	501	7	r	r	NOUN
ejpam-4412	501	8	on	on	ADP
ejpam-4412	501	9	h	h	NOUN
ejpam-4412	501	10	by	by	ADP
ejpam-4412	501	11	r	r	X
ejpam-4412	501	12	(	(	PUNCT
ejpam-4412	501	13	·	·	PUNCT
ejpam-4412	501	14	·	·	PUNCT
ejpam-4412	502	1	·	·	PUNCT
ejpam-4412	502	2	⊕	⊕	NOUN
ejpam-4412	502	3	x−2	x−2	PROPN
ejpam-4412	503	1	⊕	⊕	PROPN
ejpam-4412	503	2	x−1	x−1	PROPN
ejpam-4412	503	3	⊕	⊕	PROPN
ejpam-4412	503	4	x	x	SYM
ejpam-4412	503	5	(	(	PUNCT
ejpam-4412	503	6	0	0	NUM
ejpam-4412	503	7	)	)	PUNCT
ejpam-4412	503	8	0	0	NUM
ejpam-4412	504	1	⊕	⊕	PROPN
ejpam-4412	504	2	x1	x1	PROPN
ejpam-4412	504	3	⊕	⊕	PROPN
ejpam-4412	504	4	·	·	PUNCT
ejpam-4412	504	5	·	·	PUNCT
ejpam-4412	504	6	·	·	PUNCT
ejpam-4412	504	7	)	)	PUNCT
ejpam-4412	505	1	=	=	PUNCT
ejpam-4412	505	2	·	·	PUNCT
ejpam-4412	505	3	·	·	PUNCT
ejpam-4412	505	4	·	·	PUNCT
ejpam-4412	506	1	⊕ax−2	⊕ax−2	ADP
ejpam-4412	506	2	⊕ax	⊕ax	NOUN
ejpam-4412	506	3	(	(	PUNCT
ejpam-4412	506	4	0	0	NUM
ejpam-4412	506	5	)	)	PUNCT
ejpam-4412	506	6	−1	−1	NOUN
ejpam-4412	506	7	⊕bx0	⊕bx0	VERB
ejpam-4412	506	8	⊕bx1	⊕bx1	PROPN
ejpam-4412	506	9	⊕	⊕	PROPN
ejpam-4412	506	10	·	·	PUNCT
ejpam-4412	506	11	·	·	PUNCT
ejpam-4412	506	12	·	·	PUNCT
ejpam-4412	506	13	,	,	PUNCT
ejpam-4412	506	14	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	506	15	,	,	PUNCT
ejpam-4412	506	16	n.	n.	NOUN
ejpam-4412	506	17	h.	h.	PROPN
ejpam-4412	506	18	altaweel	altaweel	PROPN
ejpam-4412	506	19	/	/	SYM
ejpam-4412	506	20	eur	eur	PROPN
ejpam-4412	506	21	.	.	PUNCT
ejpam-4412	507	1	j.	j.	PROPN
ejpam-4412	507	2	pure	pure	PROPN
ejpam-4412	507	3	appl	appl	PROPN
ejpam-4412	507	4	.	.	PROPN
ejpam-4412	507	5	math	math	PROPN
ejpam-4412	507	6	,	,	PUNCT
ejpam-4412	507	7	15	15	NUM
ejpam-4412	507	8	(	(	PUNCT
ejpam-4412	507	9	3	3	NUM
ejpam-4412	507	10	)	)	PUNCT
ejpam-4412	507	11	(	(	PUNCT
ejpam-4412	507	12	2022	2022	NUM
ejpam-4412	507	13	)	)	PUNCT
ejpam-4412	507	14	,	,	PUNCT
ejpam-4412	507	15	1067	1067	NUM
ejpam-4412	507	16	-	-	SYM
ejpam-4412	507	17	1089	1089	NUM
ejpam-4412	507	18	1083	1083	NUM
ejpam-4412	507	19	where	where	SCONJ
ejpam-4412	507	20	a	a	DET
ejpam-4412	507	21	=	=	SYM
ejpam-4412	507	22	1	1	NUM
ejpam-4412	507	23	4	4	NUM
ejpam-4412	507	24	(	(	PUNCT
ejpam-4412	507	25	1	1	NUM
ejpam-4412	507	26	2	2	NUM
ejpam-4412	507	27	1	1	NUM
ejpam-4412	507	28	2	2	NUM
ejpam-4412	507	29	1	1	NUM
ejpam-4412	507	30	2	2	NUM
ejpam-4412	507	31	1	1	NUM
ejpam-4412	507	32	2	2	NUM
ejpam-4412	507	33	)	)	PUNCT
ejpam-4412	507	34	and	and	CCONJ
ejpam-4412	507	35	b	b	X
ejpam-4412	507	36	=	=	SYM
ejpam-4412	507	37	(	(	PUNCT
ejpam-4412	507	38	1	1	NUM
ejpam-4412	507	39	0	0	NUM
ejpam-4412	507	40	0	0	NUM
ejpam-4412	507	41	0	0	NUM
ejpam-4412	507	42	)	)	PUNCT
ejpam-4412	507	43	.	.	PUNCT
ejpam-4412	508	1	then	then	ADV
ejpam-4412	508	2	r	r	NOUN
ejpam-4412	508	3	is	be	AUX
ejpam-4412	508	4	a	a	DET
ejpam-4412	508	5	class	class	NOUN
ejpam-4412	508	6	p	p	NOUN
ejpam-4412	508	7	-	-	PUNCT
ejpam-4412	508	8	wa(s	wa(s	NUM
ejpam-4412	508	9	,	,	PUNCT
ejpam-4412	508	10	t	t	PROPN
ejpam-4412	508	11	)	)	PUNCT
ejpam-4412	508	12	.	.	PUNCT
ejpam-4412	509	1	moreover	moreover	ADV
ejpam-4412	509	2	,	,	PUNCT
ejpam-4412	509	3	ran(e	ran(e	NOUN
ejpam-4412	509	4	)	)	PUNCT
ejpam-4412	509	5	=	=	SYM
ejpam-4412	509	6	ker(r	ker(r	PROPN
ejpam-4412	509	7	)	)	PUNCT
ejpam-4412	509	8	,	,	PUNCT
ejpam-4412	509	9	e	e	X
ejpam-4412	509	10	is	be	AUX
ejpam-4412	509	11	not	not	PART
ejpam-4412	509	12	a	a	DET
ejpam-4412	509	13	self	self	NOUN
ejpam-4412	509	14	-	-	PUNCT
ejpam-4412	509	15	adjoint	adjoint	NOUN
ejpam-4412	509	16	and	and	CCONJ
ejpam-4412	509	17	ker(r	ker(r	PROPN
ejpam-4412	509	18	)	)	PUNCT
ejpam-4412	509	19	̸=	̸=	PROPN
ejpam-4412	509	20	ker(r∗	ker(r∗	PROPN
ejpam-4412	509	21	)	)	PUNCT
ejpam-4412	509	22	,	,	PUNCT
ejpam-4412	509	23	where	where	SCONJ
ejpam-4412	509	24	e	e	NOUN
ejpam-4412	509	25	is	be	AUX
ejpam-4412	509	26	the	the	DET
ejpam-4412	509	27	riesz	riesz	NOUN
ejpam-4412	509	28	idempotent	idempotent	NOUN
ejpam-4412	509	29	with	with	ADP
ejpam-4412	509	30	respect	respect	NOUN
ejpam-4412	509	31	to	to	ADP
ejpam-4412	509	32	0	0	NUM
ejpam-4412	509	33	,	,	PUNCT
ejpam-4412	509	34	see	see	VERB
ejpam-4412	509	35	[	[	X
ejpam-4412	509	36	31	31	NUM
ejpam-4412	509	37	,	,	PUNCT
ejpam-4412	509	38	example	example	NOUN
ejpam-4412	509	39	13	13	NUM
ejpam-4412	509	40	]	]	PUNCT
ejpam-4412	509	41	.	.	PUNCT
ejpam-4412	510	1	let	let	VERB
ejpam-4412	510	2	t	t	NOUN
ejpam-4412	510	3	=	=	SYM
ejpam-4412	510	4	r	r	NOUN
ejpam-4412	510	5	and	and	CCONJ
ejpam-4412	510	6	l	l	NOUN
ejpam-4412	511	1	=	=	PUNCT
ejpam-4412	511	2	p	p	X
ejpam-4412	511	3	be	be	AUX
ejpam-4412	511	4	the	the	DET
ejpam-4412	511	5	orthogonal	orthogonal	ADJ
ejpam-4412	511	6	projection	projection	NOUN
ejpam-4412	511	7	onto	onto	ADP
ejpam-4412	511	8	ker(t	ker(t	NOUN
ejpam-4412	511	9	)	)	PUNCT
ejpam-4412	511	10	.	.	PUNCT
ejpam-4412	512	1	then	then	ADV
ejpam-4412	512	2	t	t	PROPN
ejpam-4412	512	3	is	be	AUX
ejpam-4412	512	4	a	a	DET
ejpam-4412	512	5	class	class	NOUN
ejpam-4412	512	6	p	p	NOUN
ejpam-4412	512	7	-	-	PUNCT
ejpam-4412	512	8	wa(s	wa(s	NUM
ejpam-4412	512	9	,	,	PUNCT
ejpam-4412	512	10	t	t	NOUN
ejpam-4412	512	11	)	)	PUNCT
ejpam-4412	512	12	operator	operator	NOUN
ejpam-4412	512	13	and	and	CCONJ
ejpam-4412	512	14	tl	tl	PROPN
ejpam-4412	512	15	=	=	SYM
ejpam-4412	512	16	0	0	PUNCT
ejpam-4412	513	1	=	=	PUNCT
ejpam-4412	513	2	lt	lt	DET
ejpam-4412	513	3	∗	∗	NOUN
ejpam-4412	513	4	,	,	PUNCT
ejpam-4412	513	5	but	but	CCONJ
ejpam-4412	513	6	t	t	PROPN
ejpam-4412	513	7	∗l	∗l	PROPN
ejpam-4412	513	8	̸=	̸=	PROPN
ejpam-4412	513	9	lt	lt	PRON
ejpam-4412	513	10	.	.	PROPN
ejpam-4412	513	11	hence	hence	ADV
ejpam-4412	513	12	the	the	DET
ejpam-4412	513	13	kernel	kernel	PROPN
ejpam-4412	513	14	condition	condition	NOUN
ejpam-4412	513	15	ker(t	ker(t	NOUN
ejpam-4412	513	16	)	)	PUNCT
ejpam-4412	514	1	⊂	⊂	PROPN
ejpam-4412	514	2	ker(t	ker(t	NOUN
ejpam-4412	514	3	∗	∗	NOUN
ejpam-4412	514	4	)	)	PUNCT
ejpam-4412	514	5	is	be	AUX
ejpam-4412	514	6	necessary	necessary	ADJ
ejpam-4412	514	7	for	for	ADP
ejpam-4412	514	8	theorem	theorem	ADJ
ejpam-4412	514	9	10	10	NUM
ejpam-4412	514	10	.	.	PUNCT
ejpam-4412	514	11	corollary	corollary	ADJ
ejpam-4412	514	12	5	5	NUM
ejpam-4412	514	13	.	.	PUNCT
ejpam-4412	515	1	let	let	AUX
ejpam-4412	515	2	t	t	PROPN
ejpam-4412	515	3	∈	∈	PROPN
ejpam-4412	515	4	b(h	b(h	PROPN
ejpam-4412	515	5	)	)	PUNCT
ejpam-4412	515	6	be	be	VERB
ejpam-4412	515	7	a	a	DET
ejpam-4412	515	8	class	class	NOUN
ejpam-4412	515	9	p	p	NOUN
ejpam-4412	515	10	-	-	PUNCT
ejpam-4412	515	11	wa(s	wa(s	NUM
ejpam-4412	515	12	,	,	PUNCT
ejpam-4412	515	13	t	t	NOUN
ejpam-4412	515	14	)	)	PUNCT
ejpam-4412	515	15	operator	operator	NOUN
ejpam-4412	515	16	for	for	ADP
ejpam-4412	515	17	0	0	NUM
ejpam-4412	515	18	<	<	X
ejpam-4412	515	19	s	s	PROPN
ejpam-4412	515	20	,	,	PUNCT
ejpam-4412	515	21	t	t	PROPN
ejpam-4412	515	22	,	,	PUNCT
ejpam-4412	515	23	s	s	PART
ejpam-4412	515	24	+	+	NUM
ejpam-4412	515	25	t	t	X
ejpam-4412	515	26	=	=	SYM
ejpam-4412	515	27	1	1	NUM
ejpam-4412	515	28	and	and	CCONJ
ejpam-4412	515	29	0	0	NUM
ejpam-4412	516	1	<	<	X
ejpam-4412	516	2	p	p	X
ejpam-4412	516	3	≤	≤	NUM
ejpam-4412	516	4	1	1	NUM
ejpam-4412	516	5	and	and	CCONJ
ejpam-4412	516	6	ker(t	ker(t	NOUN
ejpam-4412	516	7	)	)	PUNCT
ejpam-4412	516	8	⊂	⊂	PROPN
ejpam-4412	516	9	ker(t	ker(t	NOUN
ejpam-4412	516	10	∗	∗	NOUN
ejpam-4412	516	11	)	)	PUNCT
ejpam-4412	516	12	.	.	PUNCT
ejpam-4412	517	1	if	if	SCONJ
ejpam-4412	517	2	tx	tx	PROPN
ejpam-4412	517	3	=	=	SYM
ejpam-4412	517	4	xt	xt	PROPN
ejpam-4412	517	5	∗	∗	NOUN
ejpam-4412	517	6	for	for	ADP
ejpam-4412	517	7	some	some	DET
ejpam-4412	517	8	x	x	SYM
ejpam-4412	517	9	∈	∈	PROPN
ejpam-4412	517	10	b(h	b(h	PROPN
ejpam-4412	517	11	)	)	PUNCT
ejpam-4412	517	12	then	then	ADV
ejpam-4412	517	13	t	t	X
ejpam-4412	517	14	∗x	∗x	PROPN
ejpam-4412	517	15	=	=	SYM
ejpam-4412	517	16	xt	xt	X
ejpam-4412	517	17	.	.	PUNCT
ejpam-4412	518	1	proof	proof	NOUN
ejpam-4412	518	2	.	.	PUNCT
ejpam-4412	519	1	let	let	VERB
ejpam-4412	519	2	x	x	PUNCT
ejpam-4412	519	3	=	=	PRON
ejpam-4412	519	4	l+	l+	PUNCT
ejpam-4412	519	5	ij	ij	INTJ
ejpam-4412	519	6	be	be	AUX
ejpam-4412	519	7	the	the	DET
ejpam-4412	519	8	cartesian	cartesian	ADJ
ejpam-4412	519	9	decomposition	decomposition	NOUN
ejpam-4412	519	10	of	of	ADP
ejpam-4412	519	11	x.	x.	NOUN
ejpam-4412	519	12	then	then	ADV
ejpam-4412	519	13	we	we	PRON
ejpam-4412	519	14	have	have	VERB
ejpam-4412	519	15	tl	tl	PROPN
ejpam-4412	519	16	=	=	PUNCT
ejpam-4412	519	17	lt	lt	PRON
ejpam-4412	519	18	∗	∗	NOUN
ejpam-4412	519	19	and	and	CCONJ
ejpam-4412	519	20	tj	tj	NOUN
ejpam-4412	519	21	=	=	PROPN
ejpam-4412	519	22	jt	jt	PROPN
ejpam-4412	519	23	∗	∗	NOUN
ejpam-4412	519	24	by	by	ADP
ejpam-4412	519	25	the	the	DET
ejpam-4412	519	26	assumption	assumption	NOUN
ejpam-4412	519	27	.	.	PUNCT
ejpam-4412	520	1	by	by	ADP
ejpam-4412	520	2	theorem	theorem	NOUN
ejpam-4412	520	3	10	10	NUM
ejpam-4412	520	4	,	,	PUNCT
ejpam-4412	520	5	we	we	PRON
ejpam-4412	520	6	have	have	VERB
ejpam-4412	520	7	t	t	NOUN
ejpam-4412	520	8	∗l	∗l	NOUN
ejpam-4412	520	9	=	=	SYM
ejpam-4412	520	10	lt	lt	NOUN
ejpam-4412	521	1	and	and	CCONJ
ejpam-4412	521	2	t	t	PROPN
ejpam-4412	521	3	∗j	∗j	PROPN
ejpam-4412	521	4	=	=	SYM
ejpam-4412	521	5	jt	jt	PROPN
ejpam-4412	521	6	.	.	PUNCT
ejpam-4412	522	1	this	this	PRON
ejpam-4412	522	2	implies	imply	VERB
ejpam-4412	522	3	that	that	SCONJ
ejpam-4412	522	4	t	t	VERB
ejpam-4412	522	5	∗x	∗x	PROPN
ejpam-4412	522	6	=	=	SYM
ejpam-4412	522	7	xt	xt	X
ejpam-4412	522	8	.	.	PUNCT
ejpam-4412	523	1	if	if	SCONJ
ejpam-4412	523	2	we	we	PRON
ejpam-4412	523	3	use	use	VERB
ejpam-4412	523	4	the	the	DET
ejpam-4412	523	5	2×	2×	NUM
ejpam-4412	523	6	2	2	NUM
ejpam-4412	523	7	matrix	matrix	NOUN
ejpam-4412	523	8	trick	trick	NOUN
ejpam-4412	523	9	,	,	PUNCT
ejpam-4412	523	10	we	we	PRON
ejpam-4412	523	11	easily	easily	ADV
ejpam-4412	523	12	deduce	deduce	VERB
ejpam-4412	523	13	the	the	DET
ejpam-4412	523	14	following	follow	VERB
ejpam-4412	523	15	result	result	NOUN
ejpam-4412	523	16	.	.	PUNCT
ejpam-4412	524	1	corollary	corollary	ADJ
ejpam-4412	524	2	6	6	NUM
ejpam-4412	524	3	.	.	PUNCT
ejpam-4412	524	4	suppose	suppose	VERB
ejpam-4412	524	5	that	that	SCONJ
ejpam-4412	524	6	0	0	NUM
ejpam-4412	524	7	<	<	X
ejpam-4412	524	8	s	s	PROPN
ejpam-4412	524	9	,	,	PUNCT
ejpam-4412	524	10	t	t	PROPN
ejpam-4412	524	11	,	,	PUNCT
ejpam-4412	524	12	s	s	PART
ejpam-4412	524	13	+	+	NUM
ejpam-4412	524	14	t	t	X
ejpam-4412	524	15	=	=	SYM
ejpam-4412	524	16	1	1	NUM
ejpam-4412	524	17	and	and	CCONJ
ejpam-4412	524	18	0	0	NUM
ejpam-4412	524	19	<	<	X
ejpam-4412	524	20	p	p	X
ejpam-4412	524	21	≤	≤	NUM
ejpam-4412	524	22	1	1	NUM
ejpam-4412	524	23	.	.	PUNCT
ejpam-4412	525	1	let	let	AUX
ejpam-4412	525	2	t	t	PROPN
ejpam-4412	525	3	∗	∗	NOUN
ejpam-4412	525	4	∈	∈	PROPN
ejpam-4412	525	5	b(h	b(h	PROPN
ejpam-4412	525	6	)	)	PUNCT
ejpam-4412	525	7	be	be	VERB
ejpam-4412	525	8	a	a	DET
ejpam-4412	525	9	class	class	NOUN
ejpam-4412	525	10	p	p	NOUN
ejpam-4412	525	11	-	-	PUNCT
ejpam-4412	525	12	wa(s	wa(s	NUM
ejpam-4412	525	13	,	,	PUNCT
ejpam-4412	525	14	t	t	NOUN
ejpam-4412	525	15	)	)	PUNCT
ejpam-4412	525	16	operator	operator	NOUN
ejpam-4412	525	17	and	and	CCONJ
ejpam-4412	525	18	s	s	NOUN
ejpam-4412	525	19	∈	∈	PROPN
ejpam-4412	525	20	b(k	b(k	PROPN
ejpam-4412	525	21	)	)	PUNCT
ejpam-4412	525	22	be	be	AUX
ejpam-4412	525	23	a	a	DET
ejpam-4412	525	24	class	class	NOUN
ejpam-4412	525	25	p	p	NOUN
ejpam-4412	525	26	-	-	PUNCT
ejpam-4412	525	27	wa(s	wa(s	NUM
ejpam-4412	525	28	,	,	PUNCT
ejpam-4412	525	29	t	t	NOUN
ejpam-4412	525	30	)	)	PUNCT
ejpam-4412	525	31	operator	operator	NOUN
ejpam-4412	525	32	with	with	ADP
ejpam-4412	525	33	ker(t	ker(t	NOUN
ejpam-4412	525	34	∗	∗	NOUN
ejpam-4412	525	35	)	)	PUNCT
ejpam-4412	526	1	⊂	⊂	PROPN
ejpam-4412	526	2	ker(t	ker(t	NOUN
ejpam-4412	526	3	)	)	PUNCT
ejpam-4412	526	4	and	and	CCONJ
ejpam-4412	526	5	ker(s	ker(s	NOUN
ejpam-4412	526	6	)	)	PUNCT
ejpam-4412	526	7	⊂	⊂	PRON
ejpam-4412	526	8	ker(s∗	ker(s∗	PROPN
ejpam-4412	526	9	)	)	PUNCT
ejpam-4412	526	10	.	.	PUNCT
ejpam-4412	527	1	if	if	SCONJ
ejpam-4412	527	2	x	x	PROPN
ejpam-4412	527	3	∈	∈	PROPN
ejpam-4412	527	4	b(h	b(h	PROPN
ejpam-4412	527	5	,	,	PUNCT
ejpam-4412	527	6	k	k	NOUN
ejpam-4412	527	7	)	)	PUNCT
ejpam-4412	527	8	and	and	CCONJ
ejpam-4412	527	9	xt	xt	X
ejpam-4412	527	10	=	=	SYM
ejpam-4412	527	11	sx	sx	PROPN
ejpam-4412	527	12	,	,	PUNCT
ejpam-4412	527	13	then	then	ADV
ejpam-4412	527	14	xt	xt	ADP
ejpam-4412	527	15	∗	∗	NOUN
ejpam-4412	527	16	=	=	SYM
ejpam-4412	527	17	s∗x	s∗x	NOUN
ejpam-4412	527	18	.	.	PUNCT
ejpam-4412	527	19	proof	proof	NOUN
ejpam-4412	527	20	.	.	PUNCT
ejpam-4412	528	1	put	put	VERB
ejpam-4412	528	2	a	a	DET
ejpam-4412	528	3	=	=	X
ejpam-4412	528	4	(	(	PUNCT
ejpam-4412	528	5	t	t	PROPN
ejpam-4412	528	6	∗	∗	X
ejpam-4412	528	7	0	0	NUM
ejpam-4412	528	8	0	0	NUM
ejpam-4412	528	9	s	s	PART
ejpam-4412	528	10	)	)	PUNCT
ejpam-4412	528	11	and	and	CCONJ
ejpam-4412	528	12	b	b	X
ejpam-4412	528	13	=	=	SYM
ejpam-4412	528	14	(	(	PUNCT
ejpam-4412	528	15	0	0	NUM
ejpam-4412	528	16	0	0	NUM
ejpam-4412	528	17	x	x	SYM
ejpam-4412	528	18	0	0	NUM
ejpam-4412	528	19	)	)	PUNCT
ejpam-4412	528	20	on	on	ADP
ejpam-4412	528	21	h⊕k	h⊕k	PROPN
ejpam-4412	528	22	.	.	PUNCT
ejpam-4412	529	1	then	then	ADV
ejpam-4412	529	2	a	a	PRON
ejpam-4412	529	3	is	be	AUX
ejpam-4412	529	4	a	a	DET
ejpam-4412	529	5	class	class	NOUN
ejpam-4412	529	6	p	p	NOUN
ejpam-4412	529	7	-	-	PUNCT
ejpam-4412	529	8	wa(s	wa(s	NUM
ejpam-4412	529	9	,	,	PUNCT
ejpam-4412	529	10	t	t	NOUN
ejpam-4412	529	11	)	)	PUNCT
ejpam-4412	529	12	operator	operator	NOUN
ejpam-4412	529	13	on	on	ADP
ejpam-4412	529	14	h	h	PROPN
ejpam-4412	529	15	⊕	⊕	PROPN
ejpam-4412	529	16	k	k	PROPN
ejpam-4412	529	17	that	that	SCONJ
ejpam-4412	529	18	satisfies	satisfy	VERB
ejpam-4412	529	19	ba∗	ba∗	PROPN
ejpam-4412	529	20	=	=	SYM
ejpam-4412	529	21	ab	ab	PROPN
ejpam-4412	529	22	and	and	CCONJ
ejpam-4412	529	23	ker(a	ker(a	PROPN
ejpam-4412	529	24	)	)	PUNCT
ejpam-4412	529	25	⊂	⊂	PROPN
ejpam-4412	529	26	ker(a∗	ker(a∗	X
ejpam-4412	529	27	)	)	PUNCT
ejpam-4412	529	28	.	.	PUNCT
ejpam-4412	530	1	hence	hence	ADV
ejpam-4412	530	2	we	we	PRON
ejpam-4412	530	3	have	have	VERB
ejpam-4412	530	4	ba	ba	PROPN
ejpam-4412	530	5	=	=	PUNCT
ejpam-4412	530	6	a∗b	a∗b	PROPN
ejpam-4412	530	7	,	,	PUNCT
ejpam-4412	530	8	by	by	ADP
ejpam-4412	530	9	corollary	corollary	ADJ
ejpam-4412	530	10	5	5	NUM
ejpam-4412	530	11	,	,	PUNCT
ejpam-4412	530	12	and	and	CCONJ
ejpam-4412	530	13	so	so	ADV
ejpam-4412	530	14	xt	xt	NOUN
ejpam-4412	530	15	∗	∗	X
ejpam-4412	530	16	=	=	SYM
ejpam-4412	530	17	s∗x	s∗x	PROPN
ejpam-4412	530	18	.	.	NOUN
ejpam-4412	530	19	example	example	NOUN
ejpam-4412	531	1	3	3	X
ejpam-4412	531	2	.	.	PUNCT
ejpam-4412	532	1	let	let	VERB
ejpam-4412	532	2	s	s	PRON
ejpam-4412	532	3	=	=	NOUN
ejpam-4412	532	4	t	t	PROPN
ejpam-4412	532	5	∗	∗	NOUN
ejpam-4412	532	6	=	=	SYM
ejpam-4412	532	7	r	r	NOUN
ejpam-4412	532	8	as	as	ADP
ejpam-4412	532	9	in	in	ADP
ejpam-4412	532	10	example	example	NOUN
ejpam-4412	532	11	2	2	NUM
ejpam-4412	532	12	and	and	CCONJ
ejpam-4412	532	13	x	x	SYM
ejpam-4412	533	1	=	=	X
ejpam-4412	533	2	p	p	NOUN
ejpam-4412	533	3	be	be	AUX
ejpam-4412	533	4	the	the	DET
ejpam-4412	533	5	orthogonal	orthogonal	ADJ
ejpam-4412	533	6	projection	projection	NOUN
ejpam-4412	533	7	onto	onto	ADP
ejpam-4412	533	8	ker(s	ker(s	PROPN
ejpam-4412	533	9	)	)	PUNCT
ejpam-4412	533	10	.	.	PUNCT
ejpam-4412	534	1	then	then	ADV
ejpam-4412	534	2	sx	sx	PROPN
ejpam-4412	534	3	=	=	SYM
ejpam-4412	534	4	0	0	PROPN
ejpam-4412	534	5	=	=	SYM
ejpam-4412	534	6	xt	xt	PROPN
ejpam-4412	534	7	,	,	PUNCT
ejpam-4412	534	8	but	but	CCONJ
ejpam-4412	534	9	s∗x	s∗x	PROPN
ejpam-4412	534	10	̸=	̸=	PROPN
ejpam-4412	534	11	xt	xt	ADP
ejpam-4412	535	1	∗.	∗.	PROPN
ejpam-4412	535	2	hence	hence	ADV
ejpam-4412	535	3	the	the	DET
ejpam-4412	535	4	kernel	kernel	NOUN
ejpam-4412	535	5	condition	condition	NOUN
ejpam-4412	535	6	is	be	AUX
ejpam-4412	535	7	necessary	necessary	ADJ
ejpam-4412	535	8	for	for	ADP
ejpam-4412	535	9	corollary	corollary	ADJ
ejpam-4412	535	10	6	6	NUM
ejpam-4412	535	11	.	.	PUNCT
ejpam-4412	536	1	as	as	ADP
ejpam-4412	536	2	an	an	DET
ejpam-4412	536	3	application	application	NOUN
ejpam-4412	536	4	of	of	ADP
ejpam-4412	536	5	corollary	corollary	ADJ
ejpam-4412	536	6	6	6	NUM
ejpam-4412	536	7	,	,	PUNCT
ejpam-4412	536	8	we	we	PRON
ejpam-4412	536	9	establish	establish	VERB
ejpam-4412	536	10	the	the	DET
ejpam-4412	536	11	following	follow	VERB
ejpam-4412	536	12	result	result	NOUN
ejpam-4412	536	13	.	.	PUNCT
ejpam-4412	537	1	corollary	corollary	ADJ
ejpam-4412	537	2	7	7	PROPN
ejpam-4412	537	3	.	.	PUNCT
ejpam-4412	537	4	suppose	suppose	VERB
ejpam-4412	537	5	that	that	SCONJ
ejpam-4412	537	6	0	0	NUM
ejpam-4412	537	7	<	<	X
ejpam-4412	537	8	s	s	PROPN
ejpam-4412	537	9	,	,	PUNCT
ejpam-4412	537	10	t	t	PROPN
ejpam-4412	537	11	,	,	PUNCT
ejpam-4412	537	12	s	s	PART
ejpam-4412	537	13	+	+	NUM
ejpam-4412	537	14	t	t	NOUN
ejpam-4412	537	15	=	=	SYM
ejpam-4412	537	16	1	1	X
ejpam-4412	537	17	.	.	PUNCT
ejpam-4412	538	1	let	let	VERB
ejpam-4412	538	2	t	t	PROPN
ejpam-4412	538	3	∈	∈	PROPN
ejpam-4412	538	4	b(h	b(h	PROPN
ejpam-4412	538	5	)	)	PUNCT
ejpam-4412	538	6	and	and	CCONJ
ejpam-4412	538	7	s∗	s∗	PROPN
ejpam-4412	538	8	∈	∈	PROPN
ejpam-4412	538	9	b(k	b(k	PROPN
ejpam-4412	538	10	)	)	PUNCT
ejpam-4412	538	11	be	be	AUX
ejpam-4412	538	12	class	class	NOUN
ejpam-4412	538	13	p	p	NOUN
ejpam-4412	538	14	-	-	PUNCT
ejpam-4412	538	15	wa(s	wa(s	NUM
ejpam-4412	538	16	,	,	PUNCT
ejpam-4412	538	17	t	t	NOUN
ejpam-4412	538	18	)	)	PUNCT
ejpam-4412	538	19	and	and	CCONJ
ejpam-4412	538	20	ker(t	ker(t	NOUN
ejpam-4412	538	21	)	)	PUNCT
ejpam-4412	539	1	⊂	⊂	PROPN
ejpam-4412	539	2	ker(t	ker(t	NOUN
ejpam-4412	539	3	∗	∗	NOUN
ejpam-4412	539	4	)	)	PUNCT
ejpam-4412	539	5	,	,	PUNCT
ejpam-4412	539	6	ker(s∗	ker(s∗	X
ejpam-4412	539	7	)	)	PUNCT
ejpam-4412	539	8	⊂	⊂	PROPN
ejpam-4412	540	1	ker(s	ker(s	PROPN
ejpam-4412	540	2	)	)	PUNCT
ejpam-4412	540	3	.	.	PUNCT
ejpam-4412	541	1	let	let	VERB
ejpam-4412	541	2	tx	tx	VERB
ejpam-4412	541	3	=	=	PUNCT
ejpam-4412	541	4	xs	xs	PROPN
ejpam-4412	541	5	for	for	ADP
ejpam-4412	541	6	some	some	DET
ejpam-4412	541	7	operator	operator	NOUN
ejpam-4412	541	8	x	x	SYM
ejpam-4412	541	9	∈	∈	PROPN
ejpam-4412	541	10	b(k	b(k	PROPN
ejpam-4412	541	11	,	,	PUNCT
ejpam-4412	541	12	h	h	NOUN
ejpam-4412	541	13	)	)	PUNCT
ejpam-4412	541	14	.	.	PUNCT
ejpam-4412	542	1	then	then	ADV
ejpam-4412	542	2	ran(x	ran(x	X
ejpam-4412	542	3	)	)	PUNCT
ejpam-4412	542	4	reduces	reduce	VERB
ejpam-4412	542	5	t	t	NOUN
ejpam-4412	542	6	,	,	PUNCT
ejpam-4412	542	7	ker(s)⊥	ker(s)⊥	PROPN
ejpam-4412	542	8	reduces	reduce	VERB
ejpam-4412	542	9	s	s	PRON
ejpam-4412	542	10	and	and	CCONJ
ejpam-4412	542	11	t	t	NOUN
ejpam-4412	542	12	|	|	ADV
ejpam-4412	542	13	ran(x	ran(x	PROPN
ejpam-4412	542	14	)	)	PUNCT
ejpam-4412	542	15	,	,	PUNCT
ejpam-4412	542	16	s|ker(x)⊥	s|ker(x)⊥	NOUN
ejpam-4412	542	17	are	be	AUX
ejpam-4412	542	18	unitarily	unitarily	ADV
ejpam-4412	542	19	equivalent	equivalent	ADJ
ejpam-4412	542	20	normal	normal	ADJ
ejpam-4412	542	21	operators	operator	NOUN
ejpam-4412	542	22	.	.	PUNCT
ejpam-4412	543	1	proof	proof	NOUN
ejpam-4412	543	2	.	.	PUNCT
ejpam-4412	544	1	by	by	ADP
ejpam-4412	544	2	corollary	corollary	ADJ
ejpam-4412	544	3	6	6	NUM
ejpam-4412	544	4	,	,	PUNCT
ejpam-4412	544	5	t	t	X
ejpam-4412	544	6	∗x	∗x	PROPN
ejpam-4412	544	7	=	=	SYM
ejpam-4412	544	8	xs∗.	xs∗.	PROPN
ejpam-4412	544	9	therefore	therefore	ADV
ejpam-4412	544	10	t	t	VERB
ejpam-4412	544	11	∗tx	∗tx	PUNCT
ejpam-4412	544	12	=	=	PUNCT
ejpam-4412	544	13	xs∗s	xs∗s	NOUN
ejpam-4412	544	14	and	and	CCONJ
ejpam-4412	544	15	so	so	ADV
ejpam-4412	544	16	|t	|t	PROPN
ejpam-4412	544	17	|x	|x	PROPN
ejpam-4412	544	18	=	=	PUNCT
ejpam-4412	544	19	x|s|	x|s|	PROPN
ejpam-4412	544	20	.	.	PUNCT
ejpam-4412	545	1	let	let	VERB
ejpam-4412	545	2	t	t	NOUN
ejpam-4412	545	3	=	=	SYM
ejpam-4412	545	4	u	u	NOUN
ejpam-4412	545	5	|t	|t	VERB
ejpam-4412	545	6	|	|	ADV
ejpam-4412	545	7	,	,	PUNCT
ejpam-4412	545	8	s	s	PART
ejpam-4412	545	9	=	=	SYM
ejpam-4412	545	10	v	v	NOUN
ejpam-4412	545	11	|s|	|s|	NOUN
ejpam-4412	545	12	be	be	VERB
ejpam-4412	545	13	the	the	DET
ejpam-4412	545	14	polar	polar	ADJ
ejpam-4412	545	15	decomposition	decomposition	NOUN
ejpam-4412	545	16	.	.	PUNCT
ejpam-4412	546	1	then	then	ADV
ejpam-4412	546	2	ux|s|	ux|s|	ADJ
ejpam-4412	546	3	=	=	PUNCT
ejpam-4412	546	4	u	u	SYM
ejpam-4412	546	5	|t	|t	NOUN
ejpam-4412	546	6	|x	|x	NOUN
ejpam-4412	546	7	=	=	PUNCT
ejpam-4412	546	8	tx	tx	PROPN
ejpam-4412	546	9	=	=	PUNCT
ejpam-4412	546	10	xs	xs	PROPN
ejpam-4412	546	11	=	=	PUNCT
ejpam-4412	546	12	xv	xv	PROPN
ejpam-4412	546	13	|s|	|s|	PROPN
ejpam-4412	546	14	.	.	PUNCT
ejpam-4412	547	1	let	let	VERB
ejpam-4412	547	2	x	x	SYM
ejpam-4412	547	3	∈	∈	PROPN
ejpam-4412	547	4	ker(|s|	ker(|s|	PROPN
ejpam-4412	547	5	)	)	PUNCT
ejpam-4412	547	6	.	.	PUNCT
ejpam-4412	548	1	then	then	ADV
ejpam-4412	548	2	v	v	X
ejpam-4412	548	3	x	x	SYM
ejpam-4412	548	4	=	=	SYM
ejpam-4412	548	5	0	0	NUM
ejpam-4412	548	6	and	and	CCONJ
ejpam-4412	548	7	txx	txx	X
ejpam-4412	548	8	=	=	PUNCT
ejpam-4412	548	9	xsx	xsx	X
ejpam-4412	549	1	=	=	SYM
ejpam-4412	550	1	0	0	X
ejpam-4412	550	2	.	.	PUNCT
ejpam-4412	551	1	hence	hence	ADV
ejpam-4412	551	2	xx	xx	NUM
ejpam-4412	551	3	∈	∈	NOUN
ejpam-4412	551	4	ker(t	ker(t	NOUN
ejpam-4412	551	5	)	)	PUNCT
ejpam-4412	551	6	=	=	SYM
ejpam-4412	551	7	ker(u	ker(u	PROPN
ejpam-4412	551	8	)	)	PUNCT
ejpam-4412	551	9	and	and	CCONJ
ejpam-4412	551	10	uxx	uxx	X
ejpam-4412	551	11	=	=	SYM
ejpam-4412	551	12	0	0	X
ejpam-4412	551	13	.	.	PUNCT
ejpam-4412	552	1	hence	hence	ADV
ejpam-4412	552	2	ux	ux	ADV
ejpam-4412	552	3	=	=	SYM
ejpam-4412	552	4	xv	xv	PROPN
ejpam-4412	552	5	.	.	PUNCT
ejpam-4412	553	1	since	since	SCONJ
ejpam-4412	553	2	ker(u	ker(u	PROPN
ejpam-4412	553	3	)	)	PUNCT
ejpam-4412	553	4	=	=	SYM
ejpam-4412	553	5	ker(t	ker(t	NOUN
ejpam-4412	553	6	)	)	PUNCT
ejpam-4412	554	1	⊂	⊂	PROPN
ejpam-4412	554	2	ker(t	ker(t	NOUN
ejpam-4412	554	3	∗	∗	NOUN
ejpam-4412	554	4	)	)	PUNCT
ejpam-4412	554	5	=	=	SYM
ejpam-4412	554	6	ker(u∗	ker(u∗	X
ejpam-4412	554	7	)	)	PUNCT
ejpam-4412	554	8	,	,	PUNCT
ejpam-4412	554	9	uu∗	uu∗	ADV
ejpam-4412	554	10	≤	≤	NUM
ejpam-4412	554	11	u∗u	u∗u	PUNCT
ejpam-4412	554	12	.	.	PUNCT
ejpam-4412	555	1	hence	hence	ADV
ejpam-4412	555	2	u∗uu	u∗uu	VERB
ejpam-4412	555	3	=	=	SYM
ejpam-4412	555	4	u∗uuu∗u	u∗uuu∗u	NOUN
ejpam-4412	555	5	=	=	SYM
ejpam-4412	555	6	uu∗u	uu∗u	PROPN
ejpam-4412	555	7	=	=	SYM
ejpam-4412	555	8	u	u	PROPN
ejpam-4412	555	9	.	.	PUNCT
ejpam-4412	556	1	this	this	PRON
ejpam-4412	556	2	implies	imply	VERB
ejpam-4412	556	3	u	u	NOUN
ejpam-4412	556	4	and	and	CCONJ
ejpam-4412	556	5	v	v	NOUN
ejpam-4412	556	6	∗	∗	NOUN
ejpam-4412	556	7	are	be	AUX
ejpam-4412	556	8	quasinormal	quasinormal	ADJ
ejpam-4412	556	9	.	.	PUNCT
ejpam-4412	557	1	hence	hence	ADV
ejpam-4412	557	2	u∗x	u∗x	X
ejpam-4412	557	3	=	=	SYM
ejpam-4412	557	4	xv	xv	PROPN
ejpam-4412	557	5	∗	∗	NOUN
ejpam-4412	557	6	,	,	PUNCT
ejpam-4412	557	7	ran(x	ran(x	NOUN
ejpam-4412	557	8	)	)	PUNCT
ejpam-4412	557	9	reduces	reduce	VERB
ejpam-4412	557	10	u	u	NOUN
ejpam-4412	557	11	,	,	PUNCT
ejpam-4412	557	12	|t	|t	VERB
ejpam-4412	557	13	|	|	ADV
ejpam-4412	557	14	,	,	PUNCT
ejpam-4412	557	15	ker(x)⊥	ker(x)⊥	NOUN
ejpam-4412	557	16	reduces	reduce	VERB
ejpam-4412	557	17	v	v	NOUN
ejpam-4412	557	18	,	,	PUNCT
ejpam-4412	557	19	|s|	|s|	PROPN
ejpam-4412	557	20	.	.	PUNCT
ejpam-4412	558	1	we	we	PRON
ejpam-4412	558	2	may	may	AUX
ejpam-4412	558	3	assume	assume	VERB
ejpam-4412	558	4	t	t	PROPN
ejpam-4412	558	5	<	<	X
ejpam-4412	558	6	s.	s.	PROPN
ejpam-4412	558	7	then	then	ADV
ejpam-4412	558	8	t	t	PROPN
ejpam-4412	558	9	,	,	PUNCT
ejpam-4412	558	10	s∗	s∗	PROPN
ejpam-4412	558	11	are	be	AUX
ejpam-4412	558	12	class	class	NOUN
ejpam-4412	558	13	p	p	NOUN
ejpam-4412	558	14	-	-	PUNCT
ejpam-4412	558	15	wa(s	wa(s	NUM
ejpam-4412	558	16	,	,	PUNCT
ejpam-4412	558	17	s	s	X
ejpam-4412	558	18	)	)	PUNCT
ejpam-4412	558	19	operators	operator	NOUN
ejpam-4412	558	20	with	with	ADP
ejpam-4412	558	21	reducing	reduce	VERB
ejpam-4412	558	22	kernels	kernel	NOUN
ejpam-4412	558	23	.	.	PUNCT
ejpam-4412	559	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	559	2	,	,	PUNCT
ejpam-4412	559	3	n.	n.	NOUN
ejpam-4412	559	4	h.	h.	PROPN
ejpam-4412	559	5	altaweel	altaweel	PROPN
ejpam-4412	559	6	/	/	SYM
ejpam-4412	559	7	eur	eur	PROPN
ejpam-4412	559	8	.	.	PUNCT
ejpam-4412	560	1	j.	j.	PROPN
ejpam-4412	560	2	pure	pure	PROPN
ejpam-4412	560	3	appl	appl	PROPN
ejpam-4412	560	4	.	.	PROPN
ejpam-4412	560	5	math	math	PROPN
ejpam-4412	560	6	,	,	PUNCT
ejpam-4412	560	7	15	15	NUM
ejpam-4412	560	8	(	(	PUNCT
ejpam-4412	560	9	3	3	NUM
ejpam-4412	560	10	)	)	PUNCT
ejpam-4412	560	11	(	(	PUNCT
ejpam-4412	560	12	2022	2022	NUM
ejpam-4412	560	13	)	)	PUNCT
ejpam-4412	560	14	,	,	PUNCT
ejpam-4412	560	15	1067	1067	NUM
ejpam-4412	560	16	-	-	SYM
ejpam-4412	560	17	1089	1089	NUM
ejpam-4412	560	18	1084	1084	NUM
ejpam-4412	560	19	let	let	VERB
ejpam-4412	560	20	t	t	PROPN
ejpam-4412	560	21	(	(	PUNCT
ejpam-4412	560	22	s	s	PROPN
ejpam-4412	560	23	,	,	PUNCT
ejpam-4412	560	24	s	s	PART
ejpam-4412	560	25	)	)	PUNCT
ejpam-4412	560	26	=	=	SYM
ejpam-4412	560	27	|t	|t	PROPN
ejpam-4412	560	28	|su	|su	NOUN
ejpam-4412	560	29	|t	|t	NOUN
ejpam-4412	560	30	|s	|s	PROPN
ejpam-4412	560	31	,	,	PUNCT
ejpam-4412	560	32	s(s	s(s	PROPN
ejpam-4412	560	33	,	,	PUNCT
ejpam-4412	560	34	s	s	PART
ejpam-4412	560	35	)	)	PUNCT
ejpam-4412	560	36	=	=	SYM
ejpam-4412	560	37	|s|sv	|s|sv	PROPN
ejpam-4412	560	38	|s|s	|s|s	PROPN
ejpam-4412	560	39	.	.	PUNCT
ejpam-4412	561	1	then	then	ADV
ejpam-4412	561	2	t	t	PROPN
ejpam-4412	561	3	(	(	PUNCT
ejpam-4412	561	4	s	s	PROPN
ejpam-4412	561	5	,	,	PUNCT
ejpam-4412	561	6	s	s	PART
ejpam-4412	561	7	)	)	PUNCT
ejpam-4412	561	8	,	,	PUNCT
ejpam-4412	561	9	s∗(s	s∗(s	PROPN
ejpam-4412	561	10	,	,	PUNCT
ejpam-4412	561	11	s	s	PART
ejpam-4412	561	12	)	)	PUNCT
ejpam-4412	561	13	=	=	SYM
ejpam-4412	561	14	|s∗|sv	|s∗|sv	NOUN
ejpam-4412	561	15	∗|s∗|s	∗|s∗|s	INTJ
ejpam-4412	561	16	=	=	NOUN
ejpam-4412	561	17	v	v	PROPN
ejpam-4412	561	18	s(s	s(s	PROPN
ejpam-4412	561	19	,	,	PUNCT
ejpam-4412	561	20	s)∗v	s)∗v	NOUN
ejpam-4412	561	21	∗	∗	NOUN
ejpam-4412	561	22	are	be	AUX
ejpam-4412	561	23	p	p	NOUN
ejpam-4412	561	24	2	2	NUM
ejpam-4412	561	25	-hyponormal	-hyponormal	NOUN
ejpam-4412	561	26	.	.	PUNCT
ejpam-4412	562	1	also	also	ADV
ejpam-4412	562	2	,	,	PUNCT
ejpam-4412	562	3	since	since	SCONJ
ejpam-4412	562	4	|s(s	|s(s	PROPN
ejpam-4412	562	5	,	,	PUNCT
ejpam-4412	562	6	s)∗|	s)∗|	NOUN
ejpam-4412	563	1	−	−	PROPN
ejpam-4412	564	1	|s(s	|s(s	PROPN
ejpam-4412	564	2	,	,	PUNCT
ejpam-4412	564	3	s)|	s)|	NOUN
ejpam-4412	564	4	=	=	NOUN
ejpam-4412	564	5	v	v	NOUN
ejpam-4412	564	6	∗(|s∗(s	∗(|s∗(s	NOUN
ejpam-4412	564	7	,	,	PUNCT
ejpam-4412	564	8	s)|	s)|	NOUN
ejpam-4412	564	9	−	−	PROPN
ejpam-4412	564	10	|s∗(s	|s∗(s	NOUN
ejpam-4412	564	11	,	,	PUNCT
ejpam-4412	564	12	s)∗|)v	s)∗|)v	VERB
ejpam-4412	564	13	≥	≥	NOUN
ejpam-4412	564	14	0	0	NUM
ejpam-4412	564	15	,	,	PUNCT
ejpam-4412	564	16	s(s	s(s	PROPN
ejpam-4412	564	17	,	,	PUNCT
ejpam-4412	564	18	s)∗	s)∗	X
ejpam-4412	564	19	is	be	AUX
ejpam-4412	564	20	p	p	X
ejpam-4412	564	21	2	2	NUM
ejpam-4412	564	22	-hyponormal	-hyponormal	NOUN
ejpam-4412	564	23	,	,	PUNCT
ejpam-4412	564	24	too	too	ADV
ejpam-4412	564	25	.	.	PUNCT
ejpam-4412	565	1	then	then	ADV
ejpam-4412	565	2	t	t	PROPN
ejpam-4412	565	3	(	(	PUNCT
ejpam-4412	565	4	s	s	X
ejpam-4412	565	5	,	,	PUNCT
ejpam-4412	565	6	s)x	s)x	X
ejpam-4412	565	7	=	=	SYM
ejpam-4412	565	8	|t	|t	PROPN
ejpam-4412	565	9	|su	|su	X
ejpam-4412	565	10	|t	|t	VERB
ejpam-4412	565	11	|sx	|sx	PROPN
ejpam-4412	565	12	=	=	ADJ
ejpam-4412	565	13	|t	|t	NOUN
ejpam-4412	565	14	|sux|s|s	|sux|s|s	PROPN
ejpam-4412	565	15	=	=	SYM
ejpam-4412	565	16	|t	|t	PROPN
ejpam-4412	565	17	|sxv	|sxv	VERB
ejpam-4412	565	18	|s|s	|s|s	NOUN
ejpam-4412	565	19	=	=	PUNCT
ejpam-4412	565	20	xs(s	xs(s	X
ejpam-4412	565	21	,	,	PUNCT
ejpam-4412	565	22	s	s	NOUN
ejpam-4412	565	23	)	)	PUNCT
ejpam-4412	565	24	,	,	PUNCT
ejpam-4412	565	25	hence	hence	ADV
ejpam-4412	565	26	t	t	PROPN
ejpam-4412	565	27	(	(	PUNCT
ejpam-4412	565	28	s	s	PROPN
ejpam-4412	565	29	,	,	PUNCT
ejpam-4412	565	30	s)∗x	s)∗x	X
ejpam-4412	565	31	=	=	SYM
ejpam-4412	565	32	xs(s	xs(s	X
ejpam-4412	565	33	,	,	PUNCT
ejpam-4412	565	34	s)∗	s)∗	ADJ
ejpam-4412	565	35	,	,	PUNCT
ejpam-4412	565	36	ran(x	ran(x	PROPN
ejpam-4412	565	37	)	)	PUNCT
ejpam-4412	565	38	reduces	reduce	VERB
ejpam-4412	565	39	t	t	NOUN
ejpam-4412	565	40	(	(	PUNCT
ejpam-4412	565	41	s	s	PROPN
ejpam-4412	565	42	,	,	PUNCT
ejpam-4412	565	43	s	s	PART
ejpam-4412	565	44	)	)	PUNCT
ejpam-4412	565	45	,	,	PUNCT
ejpam-4412	565	46	ker(x)⊥	ker(x)⊥	PROPN
ejpam-4412	565	47	reduces	reduce	VERB
ejpam-4412	565	48	s(s	s(s	PROPN
ejpam-4412	565	49	,	,	PUNCT
ejpam-4412	565	50	s	s	PART
ejpam-4412	565	51	)	)	PUNCT
ejpam-4412	565	52	and	and	CCONJ
ejpam-4412	565	53	t	t	PROPN
ejpam-4412	565	54	|	|	ADV
ejpam-4412	565	55	ran(x	ran(x	PROPN
ejpam-4412	565	56	)	)	PUNCT
ejpam-4412	565	57	(	(	PUNCT
ejpam-4412	565	58	s	s	X
ejpam-4412	565	59	,	,	PUNCT
ejpam-4412	565	60	s	s	PART
ejpam-4412	565	61	)	)	PUNCT
ejpam-4412	565	62	=	=	SYM
ejpam-4412	565	63	t	t	PROPN
ejpam-4412	565	64	(	(	PUNCT
ejpam-4412	565	65	s	s	PROPN
ejpam-4412	565	66	,	,	PUNCT
ejpam-4412	565	67	s)|	s)|	NOUN
ejpam-4412	565	68	ran(x	ran(x	PROPN
ejpam-4412	565	69	)	)	PUNCT
ejpam-4412	565	70	≃	≃	PROPN
ejpam-4412	565	71	s(s	s(s	PROPN
ejpam-4412	565	72	,	,	PUNCT
ejpam-4412	565	73	s)|ker(x)⊥	s)|ker(x)⊥	PROPN
ejpam-4412	565	74	=	=	SYM
ejpam-4412	565	75	s|ker(x)⊥(s	s|ker(x)⊥(s	PROPN
ejpam-4412	565	76	,	,	PUNCT
ejpam-4412	565	77	s	s	PART
ejpam-4412	565	78	)	)	PUNCT
ejpam-4412	565	79	are	be	AUX
ejpam-4412	565	80	unitarily	unitarily	ADV
ejpam-4412	565	81	equivalent	equivalent	ADJ
ejpam-4412	565	82	normal	normal	ADJ
ejpam-4412	565	83	operators	operator	NOUN
ejpam-4412	565	84	.	.	PUNCT
ejpam-4412	566	1	hence	hence	ADV
ejpam-4412	566	2	t	t	PROPN
ejpam-4412	566	3	|	|	ADV
ejpam-4412	566	4	ran(x	ran(x	PROPN
ejpam-4412	566	5	)	)	PUNCT
ejpam-4412	567	1	,	,	PUNCT
ejpam-4412	567	2	s|ker(x)⊥	s|ker(x)⊥	NOUN
ejpam-4412	567	3	are	be	AUX
ejpam-4412	567	4	normal	normal	ADJ
ejpam-4412	567	5	by	by	ADP
ejpam-4412	567	6	corollary	corollary	ADJ
ejpam-4412	567	7	1	1	NUM
ejpam-4412	567	8	,	,	PUNCT
ejpam-4412	567	9	and	and	CCONJ
ejpam-4412	567	10	that	that	SCONJ
ejpam-4412	567	11	they	they	PRON
ejpam-4412	567	12	are	be	AUX
ejpam-4412	567	13	unitarily	unitarily	ADV
ejpam-4412	567	14	equivalent	equivalent	ADJ
ejpam-4412	567	15	follows	follow	VERB
ejpam-4412	567	16	from	from	ADP
ejpam-4412	567	17	the	the	DET
ejpam-4412	567	18	fact	fact	NOUN
ejpam-4412	567	19	that	that	SCONJ
ejpam-4412	567	20	if	if	SCONJ
ejpam-4412	567	21	n	n	NOUN
ejpam-4412	567	22	=	=	SYM
ejpam-4412	567	23	u	u	PROPN
ejpam-4412	567	24	|n	|n	NOUN
ejpam-4412	567	25	|	|	ADV
ejpam-4412	567	26	and	and	CCONJ
ejpam-4412	567	27	m	m	PROPN
ejpam-4412	567	28	=	=	NOUN
ejpam-4412	567	29	w	w	ADJ
ejpam-4412	567	30	|m	|m	NOUN
ejpam-4412	567	31	|	|	ADV
ejpam-4412	567	32	are	be	AUX
ejpam-4412	567	33	normal	normal	ADJ
ejpam-4412	567	34	operators	operator	NOUN
ejpam-4412	567	35	,	,	PUNCT
ejpam-4412	567	36	then	then	ADV
ejpam-4412	567	37	for	for	ADP
ejpam-4412	567	38	a	a	DET
ejpam-4412	567	39	unitary	unitary	ADJ
ejpam-4412	567	40	operator	operator	NOUN
ejpam-4412	567	41	v	v	NOUN
ejpam-4412	567	42	,	,	PUNCT
ejpam-4412	567	43	n	n	NOUN
ejpam-4412	567	44	=	=	SYM
ejpam-4412	567	45	v	v	ADP
ejpam-4412	567	46	∗mv	∗mv	PUNCT
ejpam-4412	567	47	if	if	SCONJ
ejpam-4412	567	48	and	and	CCONJ
ejpam-4412	567	49	only	only	ADV
ejpam-4412	567	50	if	if	SCONJ
ejpam-4412	567	51	u	u	PROPN
ejpam-4412	567	52	=	=	SYM
ejpam-4412	567	53	v	v	PART
ejpam-4412	567	54	∗wv	∗wv	PUNCT
ejpam-4412	567	55	and	and	CCONJ
ejpam-4412	567	56	|n	|n	ADJ
ejpam-4412	567	57	|s	|s	PROPN
ejpam-4412	567	58	=	=	SYM
ejpam-4412	567	59	v	v	PROPN
ejpam-4412	567	60	∗|m	∗|m	PUNCT
ejpam-4412	567	61	|sv	|sv	PUNCT
ejpam-4412	567	62	for	for	ADP
ejpam-4412	567	63	any	any	DET
ejpam-4412	567	64	s	s	X
ejpam-4412	567	65	>	>	X
ejpam-4412	567	66	0	0	X
ejpam-4412	567	67	.	.	PUNCT
ejpam-4412	567	68	theorem	theorem	NOUN
ejpam-4412	567	69	11	11	NUM
ejpam-4412	567	70	.	.	PUNCT
ejpam-4412	567	71	suppose	suppose	VERB
ejpam-4412	567	72	that	that	SCONJ
ejpam-4412	567	73	0	0	NUM
ejpam-4412	567	74	<	<	X
ejpam-4412	567	75	s	s	PROPN
ejpam-4412	567	76	,	,	PUNCT
ejpam-4412	567	77	t	t	PROPN
ejpam-4412	567	78	,	,	PUNCT
ejpam-4412	567	79	s	s	PART
ejpam-4412	567	80	+	+	NUM
ejpam-4412	567	81	t	t	NOUN
ejpam-4412	567	82	=	=	SYM
ejpam-4412	567	83	1	1	X
ejpam-4412	567	84	.	.	PUNCT
ejpam-4412	568	1	let	let	AUX
ejpam-4412	568	2	t	t	PROPN
ejpam-4412	568	3	∈	∈	PROPN
ejpam-4412	568	4	b(h	b(h	PROPN
ejpam-4412	568	5	)	)	PUNCT
ejpam-4412	568	6	be	be	AUX
ejpam-4412	568	7	class	class	NOUN
ejpam-4412	568	8	p	p	NOUN
ejpam-4412	568	9	-	-	PUNCT
ejpam-4412	568	10	wa(s	wa(s	NUM
ejpam-4412	568	11	,	,	PUNCT
ejpam-4412	568	12	t	t	PROPN
ejpam-4412	568	13	)	)	PUNCT
ejpam-4412	568	14	and	and	CCONJ
ejpam-4412	568	15	n	n	DET
ejpam-4412	568	16	a	a	DET
ejpam-4412	568	17	normal	normal	ADJ
ejpam-4412	568	18	operator	operator	NOUN
ejpam-4412	568	19	.	.	PUNCT
ejpam-4412	569	1	let	let	VERB
ejpam-4412	569	2	tx	tx	VERB
ejpam-4412	569	3	=	=	PUNCT
ejpam-4412	570	1	xn	xn	PROPN
ejpam-4412	570	2	.	.	PUNCT
ejpam-4412	571	1	then	then	ADV
ejpam-4412	571	2	the	the	DET
ejpam-4412	571	3	following	follow	VERB
ejpam-4412	571	4	assertions	assertion	NOUN
ejpam-4412	571	5	hold	hold	VERB
ejpam-4412	571	6	.	.	PUNCT
ejpam-4412	572	1	(	(	PUNCT
ejpam-4412	572	2	i	i	NOUN
ejpam-4412	572	3	)	)	PUNCT
ejpam-4412	572	4	if	if	SCONJ
ejpam-4412	572	5	the	the	DET
ejpam-4412	572	6	range	range	NOUN
ejpam-4412	572	7	ran(x	ran(x	X
ejpam-4412	572	8	)	)	PUNCT
ejpam-4412	572	9	is	be	AUX
ejpam-4412	572	10	dense	dense	ADJ
ejpam-4412	572	11	,	,	PUNCT
ejpam-4412	572	12	then	then	ADV
ejpam-4412	572	13	t	t	PROPN
ejpam-4412	572	14	is	be	AUX
ejpam-4412	572	15	normal	normal	ADJ
ejpam-4412	572	16	.	.	PUNCT
ejpam-4412	573	1	(	(	PUNCT
ejpam-4412	573	2	ii	ii	NOUN
ejpam-4412	573	3	)	)	PUNCT
ejpam-4412	573	4	if	if	SCONJ
ejpam-4412	573	5	ker(x∗	ker(x∗	NOUN
ejpam-4412	573	6	)	)	PUNCT
ejpam-4412	573	7	⊂	⊂	PROPN
ejpam-4412	574	1	ker(t	ker(t	NOUN
ejpam-4412	574	2	∗	∗	NOUN
ejpam-4412	574	3	)	)	PUNCT
ejpam-4412	574	4	,	,	PUNCT
ejpam-4412	574	5	then	then	ADV
ejpam-4412	574	6	t	t	PROPN
ejpam-4412	574	7	is	be	AUX
ejpam-4412	574	8	quasinormal	quasinormal	ADJ
ejpam-4412	574	9	.	.	PUNCT
ejpam-4412	575	1	proof	proof	NOUN
ejpam-4412	575	2	.	.	PUNCT
ejpam-4412	576	1	let	let	VERB
ejpam-4412	576	2	z	z	NOUN
ejpam-4412	576	3	=	=	PUNCT
ejpam-4412	576	4	|t	|t	PROPN
ejpam-4412	577	1	|sx	|sx	NUM
ejpam-4412	577	2	.	.	PUNCT
ejpam-4412	577	3	then	then	ADV
ejpam-4412	577	4	t	t	PROPN
ejpam-4412	577	5	(	(	PUNCT
ejpam-4412	577	6	s	s	X
ejpam-4412	577	7	,	,	PUNCT
ejpam-4412	577	8	t)z	t)z	NOUN
ejpam-4412	577	9	=	=	SYM
ejpam-4412	577	10	|t	|t	PROPN
ejpam-4412	577	11	|su	|su	NOUN
ejpam-4412	577	12	|t	|t	VERB
ejpam-4412	577	13	|t|t	|t|t	ADJ
ejpam-4412	577	14	|sx	|sx	PROPN
ejpam-4412	577	15	=	=	PUNCT
ejpam-4412	577	16	|t	|t	NOUN
ejpam-4412	577	17	|stx	|stx	PROPN
ejpam-4412	577	18	=	=	SYM
ejpam-4412	577	19	|t	|t	NOUN
ejpam-4412	577	20	|sxn	|sxn	NOUN
ejpam-4412	577	21	=	=	SYM
ejpam-4412	578	1	zn	zn	X
ejpam-4412	578	2	.	.	PUNCT
ejpam-4412	579	1	since	since	SCONJ
ejpam-4412	579	2	t	t	PROPN
ejpam-4412	579	3	(	(	PUNCT
ejpam-4412	579	4	s	s	PROPN
ejpam-4412	579	5	,	,	PUNCT
ejpam-4412	579	6	t	t	PROPN
ejpam-4412	579	7	)	)	PUNCT
ejpam-4412	579	8	is	be	AUX
ejpam-4412	579	9	min{sp	min{sp	ADV
ejpam-4412	579	10	,	,	PUNCT
ejpam-4412	579	11	tp}-hyponormal	tp}-hyponormal	ADJ
ejpam-4412	579	12	,	,	PUNCT
ejpam-4412	579	13	we	we	PRON
ejpam-4412	579	14	have	have	VERB
ejpam-4412	579	15	t	t	PROPN
ejpam-4412	579	16	(	(	PUNCT
ejpam-4412	579	17	s	s	X
ejpam-4412	579	18	,	,	PUNCT
ejpam-4412	579	19	t)∗z	t)∗z	NOUN
ejpam-4412	579	20	=	=	PUNCT
ejpam-4412	579	21	zn∗	zn∗	NOUN
ejpam-4412	579	22	by	by	ADP
ejpam-4412	579	23	[	[	X
ejpam-4412	579	24	30	30	NUM
ejpam-4412	579	25	]	]	PUNCT
ejpam-4412	579	26	.	.	PUNCT
ejpam-4412	580	1	hence	hence	ADV
ejpam-4412	580	2	(	(	PUNCT
ejpam-4412	580	3	t	t	PROPN
ejpam-4412	580	4	(	(	PUNCT
ejpam-4412	580	5	s	s	X
ejpam-4412	580	6	,	,	PUNCT
ejpam-4412	580	7	t)∗t	t)∗t	X
ejpam-4412	580	8	(	(	PUNCT
ejpam-4412	580	9	s	s	X
ejpam-4412	580	10	,	,	PUNCT
ejpam-4412	580	11	t)−	t)−	PROPN
ejpam-4412	580	12	t	t	PROPN
ejpam-4412	580	13	(	(	PUNCT
ejpam-4412	580	14	s	s	X
ejpam-4412	580	15	,	,	PUNCT
ejpam-4412	580	16	t)t	t)t	X
ejpam-4412	580	17	(	(	PUNCT
ejpam-4412	580	18	s	s	X
ejpam-4412	580	19	,	,	PUNCT
ejpam-4412	580	20	t)∗)|t	t)∗)|t	PROPN
ejpam-4412	580	21	|sx	|sx	PROPN
ejpam-4412	580	22	=	=	SYM
ejpam-4412	580	23	t	t	PROPN
ejpam-4412	580	24	(	(	PUNCT
ejpam-4412	580	25	s	s	X
ejpam-4412	580	26	,	,	PUNCT
ejpam-4412	580	27	t)∗t	t)∗t	X
ejpam-4412	580	28	(	(	PUNCT
ejpam-4412	580	29	s	s	NOUN
ejpam-4412	580	30	,	,	PUNCT
ejpam-4412	580	31	t)z	t)z	NOUN
ejpam-4412	580	32	−	−	PROPN
ejpam-4412	580	33	t	t	NOUN
ejpam-4412	580	34	(	(	PUNCT
ejpam-4412	580	35	s	s	X
ejpam-4412	580	36	,	,	PUNCT
ejpam-4412	580	37	t)t	t)t	X
ejpam-4412	580	38	(	(	PUNCT
ejpam-4412	580	39	s	s	X
ejpam-4412	580	40	,	,	PUNCT
ejpam-4412	580	41	t)∗z	t)∗z	PROPN
ejpam-4412	580	42	=	=	SYM
ejpam-4412	580	43	t	t	PROPN
ejpam-4412	580	44	(	(	PUNCT
ejpam-4412	580	45	s	s	X
ejpam-4412	580	46	,	,	PUNCT
ejpam-4412	580	47	t)∗zn	t)∗zn	PROPN
ejpam-4412	580	48	−	−	PROPN
ejpam-4412	580	49	t	t	PROPN
ejpam-4412	580	50	(	(	PUNCT
ejpam-4412	580	51	s	s	PROPN
ejpam-4412	580	52	,	,	PUNCT
ejpam-4412	580	53	t)zn∗	t)zn∗	NOUN
ejpam-4412	580	54	=	=	NOUN
ejpam-4412	581	1	zn∗n	zn∗n	NUM
ejpam-4412	581	2	−	−	NOUN
ejpam-4412	581	3	znn∗	znn∗	NOUN
ejpam-4412	581	4	=	=	SYM
ejpam-4412	581	5	0	0	PROPN
ejpam-4412	581	6	.	.	PUNCT
ejpam-4412	582	1	(	(	PUNCT
ejpam-4412	582	2	i	i	NOUN
ejpam-4412	582	3	)	)	PUNCT
ejpam-4412	582	4	if	if	SCONJ
ejpam-4412	582	5	ran(x	ran(x	PROPN
ejpam-4412	582	6	)	)	PUNCT
ejpam-4412	582	7	is	be	AUX
ejpam-4412	582	8	dense	dense	ADJ
ejpam-4412	582	9	,	,	PUNCT
ejpam-4412	582	10	then	then	ADV
ejpam-4412	582	11	(	(	PUNCT
ejpam-4412	582	12	t	t	PROPN
ejpam-4412	582	13	(	(	PUNCT
ejpam-4412	582	14	s	s	X
ejpam-4412	582	15	,	,	PUNCT
ejpam-4412	582	16	t)∗t	t)∗t	X
ejpam-4412	582	17	(	(	PUNCT
ejpam-4412	582	18	s	s	X
ejpam-4412	582	19	,	,	PUNCT
ejpam-4412	582	20	t)−	t)−	PROPN
ejpam-4412	582	21	t	t	PROPN
ejpam-4412	582	22	(	(	PUNCT
ejpam-4412	582	23	s	s	X
ejpam-4412	582	24	,	,	PUNCT
ejpam-4412	582	25	t)t	t)t	X
ejpam-4412	582	26	(	(	PUNCT
ejpam-4412	582	27	s	s	X
ejpam-4412	582	28	,	,	PUNCT
ejpam-4412	582	29	t)∗)|t	t)∗)|t	PROPN
ejpam-4412	582	30	|s	|s	PROPN
ejpam-4412	582	31	=	=	PROPN
ejpam-4412	583	1	0	0	X
ejpam-4412	583	2	.	.	PUNCT
ejpam-4412	584	1	since	since	SCONJ
ejpam-4412	584	2	ker(|t	ker(|t	PROPN
ejpam-4412	584	3	|s	|s	PROPN
ejpam-4412	584	4	)	)	PUNCT
ejpam-4412	584	5	⊂	⊂	PROPN
ejpam-4412	584	6	ker(t	ker(t	X
ejpam-4412	584	7	(	(	PUNCT
ejpam-4412	584	8	s	s	PROPN
ejpam-4412	584	9	,	,	PUNCT
ejpam-4412	584	10	t	t	PROPN
ejpam-4412	584	11	)	)	PUNCT
ejpam-4412	584	12	)	)	PUNCT
ejpam-4412	585	1	∩	∩	ADJ
ejpam-4412	585	2	ker(t	ker(t	NOUN
ejpam-4412	585	3	(	(	PUNCT
ejpam-4412	585	4	s	s	PROPN
ejpam-4412	585	5	,	,	PUNCT
ejpam-4412	585	6	t)∗	t)∗	NOUN
ejpam-4412	585	7	)	)	PUNCT
ejpam-4412	585	8	,	,	PUNCT
ejpam-4412	585	9	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	585	10	,	,	PUNCT
ejpam-4412	585	11	n.	n.	NOUN
ejpam-4412	585	12	h.	h.	PROPN
ejpam-4412	585	13	altaweel	altaweel	PROPN
ejpam-4412	585	14	/	/	SYM
ejpam-4412	585	15	eur	eur	PROPN
ejpam-4412	585	16	.	.	PUNCT
ejpam-4412	586	1	j.	j.	PROPN
ejpam-4412	586	2	pure	pure	PROPN
ejpam-4412	586	3	appl	appl	PROPN
ejpam-4412	586	4	.	.	PROPN
ejpam-4412	586	5	math	math	PROPN
ejpam-4412	586	6	,	,	PUNCT
ejpam-4412	586	7	15	15	NUM
ejpam-4412	586	8	(	(	PUNCT
ejpam-4412	586	9	3	3	NUM
ejpam-4412	586	10	)	)	PUNCT
ejpam-4412	586	11	(	(	PUNCT
ejpam-4412	586	12	2022	2022	NUM
ejpam-4412	586	13	)	)	PUNCT
ejpam-4412	586	14	,	,	PUNCT
ejpam-4412	586	15	1067	1067	NUM
ejpam-4412	586	16	-	-	SYM
ejpam-4412	586	17	1089	1089	NUM
ejpam-4412	586	18	1085	1085	NUM
ejpam-4412	586	19	this	this	PRON
ejpam-4412	586	20	implies	imply	VERB
ejpam-4412	586	21	t	t	PROPN
ejpam-4412	586	22	(	(	PUNCT
ejpam-4412	586	23	s	s	PROPN
ejpam-4412	586	24	,	,	PUNCT
ejpam-4412	586	25	t	t	PROPN
ejpam-4412	586	26	)	)	PUNCT
ejpam-4412	586	27	is	be	AUX
ejpam-4412	586	28	normal	normal	ADJ
ejpam-4412	586	29	.	.	PUNCT
ejpam-4412	587	1	hence	hence	ADV
ejpam-4412	587	2	t	t	PROPN
ejpam-4412	587	3	is	be	AUX
ejpam-4412	587	4	normal	normal	ADJ
ejpam-4412	587	5	by	by	ADP
ejpam-4412	587	6	corollary	corollary	ADJ
ejpam-4412	587	7	1	1	NUM
ejpam-4412	587	8	.	.	PUNCT
ejpam-4412	588	1	(	(	PUNCT
ejpam-4412	588	2	ii	ii	NOUN
ejpam-4412	588	3	)	)	PUNCT
ejpam-4412	588	4	let	let	VERB
ejpam-4412	588	5	x∗|t	x∗|t	PROPN
ejpam-4412	588	6	|sx	|sx	X
ejpam-4412	588	7	=	=	SYM
ejpam-4412	588	8	0	0	PROPN
ejpam-4412	588	9	.	.	PUNCT
ejpam-4412	589	1	then	then	ADV
ejpam-4412	589	2	|t	|t	VERB
ejpam-4412	589	3	|sx	|sx	PROPN
ejpam-4412	589	4	∈	∈	PROPN
ejpam-4412	589	5	ker(x∗	ker(x∗	PROPN
ejpam-4412	589	6	)	)	PUNCT
ejpam-4412	590	1	⊂	⊂	PROPN
ejpam-4412	591	1	ker(t	ker(t	NOUN
ejpam-4412	591	2	∗	∗	NOUN
ejpam-4412	591	3	)	)	PUNCT
ejpam-4412	591	4	=	=	SYM
ejpam-4412	591	5	ker(u∗	ker(u∗	X
ejpam-4412	591	6	)	)	PUNCT
ejpam-4412	591	7	and	and	CCONJ
ejpam-4412	591	8	t	t	PROPN
ejpam-4412	591	9	(	(	PUNCT
ejpam-4412	591	10	s	s	PROPN
ejpam-4412	591	11	,	,	PUNCT
ejpam-4412	591	12	t)∗x	t)∗x	PROPN
ejpam-4412	591	13	=	=	SYM
ejpam-4412	591	14	|t	|t	VERB
ejpam-4412	591	15	|tu∗|t	|tu∗|t	NOUN
ejpam-4412	591	16	|sx	|sx	NOUN
ejpam-4412	592	1	=	=	SYM
ejpam-4412	592	2	0	0	X
ejpam-4412	592	3	.	.	PUNCT
ejpam-4412	593	1	hence	hence	ADV
ejpam-4412	593	2	ker(x∗|t	ker(x∗|t	PROPN
ejpam-4412	593	3	|s	|s	PROPN
ejpam-4412	593	4	)	)	PUNCT
ejpam-4412	594	1	⊂	⊂	PROPN
ejpam-4412	594	2	ker(t	ker(t	X
ejpam-4412	594	3	(	(	PUNCT
ejpam-4412	594	4	s	s	PROPN
ejpam-4412	594	5	,	,	PUNCT
ejpam-4412	594	6	t)∗	t)∗	NOUN
ejpam-4412	594	7	)	)	PUNCT
ejpam-4412	594	8	and	and	CCONJ
ejpam-4412	594	9	ran(t	ran(t	PROPN
ejpam-4412	594	10	(	(	PUNCT
ejpam-4412	594	11	s	s	PROPN
ejpam-4412	594	12	,	,	PUNCT
ejpam-4412	594	13	t	t	PROPN
ejpam-4412	594	14	)	)	PUNCT
ejpam-4412	594	15	)	)	PUNCT
ejpam-4412	595	1	⊂	⊂	PROPN
ejpam-4412	596	1	ran(|t	ran(|t	ADP
ejpam-4412	596	2	|sx	|sx	X
ejpam-4412	596	3	)	)	PUNCT
ejpam-4412	596	4	.	.	PUNCT
ejpam-4412	597	1	hence	hence	ADV
ejpam-4412	597	2	(	(	PUNCT
ejpam-4412	597	3	t	t	PROPN
ejpam-4412	597	4	(	(	PUNCT
ejpam-4412	597	5	s	s	X
ejpam-4412	597	6	,	,	PUNCT
ejpam-4412	597	7	t)∗t	t)∗t	X
ejpam-4412	597	8	(	(	PUNCT
ejpam-4412	597	9	s	s	X
ejpam-4412	597	10	,	,	PUNCT
ejpam-4412	597	11	t)−	t)−	PROPN
ejpam-4412	597	12	t	t	PROPN
ejpam-4412	597	13	(	(	PUNCT
ejpam-4412	597	14	s	s	X
ejpam-4412	597	15	,	,	PUNCT
ejpam-4412	597	16	t)t	t)t	X
ejpam-4412	597	17	(	(	PUNCT
ejpam-4412	597	18	s	s	X
ejpam-4412	597	19	,	,	PUNCT
ejpam-4412	597	20	t)∗)t	t)∗)t	X
ejpam-4412	597	21	(	(	PUNCT
ejpam-4412	597	22	s	s	X
ejpam-4412	597	23	,	,	PUNCT
ejpam-4412	597	24	t	t	PROPN
ejpam-4412	597	25	)	)	PUNCT
ejpam-4412	597	26	=	=	SYM
ejpam-4412	597	27	0	0	NUM
ejpam-4412	597	28	by	by	ADP
ejpam-4412	597	29	(	(	PUNCT
ejpam-4412	597	30	i	i	NOUN
ejpam-4412	597	31	)	)	PUNCT
ejpam-4412	597	32	.	.	PUNCT
ejpam-4412	598	1	this	this	PRON
ejpam-4412	598	2	implies	imply	VERB
ejpam-4412	598	3	t	t	PROPN
ejpam-4412	598	4	(	(	PUNCT
ejpam-4412	598	5	s	s	PROPN
ejpam-4412	598	6	,	,	PUNCT
ejpam-4412	598	7	t	t	PROPN
ejpam-4412	598	8	)	)	PUNCT
ejpam-4412	598	9	is	be	AUX
ejpam-4412	598	10	quasinormal	quasinormal	ADJ
ejpam-4412	598	11	,	,	PUNCT
ejpam-4412	598	12	and	and	CCONJ
ejpam-4412	598	13	t	t	PROPN
ejpam-4412	598	14	is	be	AUX
ejpam-4412	598	15	quasinormal	quasinormal	ADJ
ejpam-4412	598	16	by	by	ADP
ejpam-4412	598	17	theorem	theorem	NOUN
ejpam-4412	598	18	1	1	NUM
ejpam-4412	598	19	.	.	PUNCT
ejpam-4412	598	20	theorem	theorem	NOUN
ejpam-4412	598	21	12	12	NUM
ejpam-4412	598	22	.	.	PUNCT
ejpam-4412	599	1	suppose	suppose	VERB
ejpam-4412	599	2	that	that	SCONJ
ejpam-4412	599	3	0	0	NUM
ejpam-4412	599	4	<	<	X
ejpam-4412	599	5	s	s	PROPN
ejpam-4412	599	6	,	,	PUNCT
ejpam-4412	599	7	t	t	PROPN
ejpam-4412	599	8	,	,	PUNCT
ejpam-4412	599	9	s	s	PART
ejpam-4412	599	10	+	+	NUM
ejpam-4412	599	11	t	t	X
ejpam-4412	599	12	=	=	SYM
ejpam-4412	599	13	1	1	NUM
ejpam-4412	599	14	and	and	CCONJ
ejpam-4412	599	15	0	0	NUM
ejpam-4412	599	16	<	<	X
ejpam-4412	599	17	q	q	X
ejpam-4412	599	18	≤	≤	NUM
ejpam-4412	599	19	1	1	NUM
ejpam-4412	599	20	.	.	PUNCT
ejpam-4412	600	1	let	let	AUX
ejpam-4412	600	2	t	t	PROPN
ejpam-4412	600	3	∈	∈	PROPN
ejpam-4412	600	4	b(h	b(h	PROPN
ejpam-4412	600	5	)	)	PUNCT
ejpam-4412	600	6	be	be	VERB
ejpam-4412	600	7	such	such	ADJ
ejpam-4412	600	8	that	that	SCONJ
ejpam-4412	600	9	t	t	PROPN
ejpam-4412	600	10	∗	∗	NOUN
ejpam-4412	600	11	is	be	AUX
ejpam-4412	600	12	p	p	ADJ
ejpam-4412	600	13	-	-	PUNCT
ejpam-4412	600	14	hyponormal	hyponormal	ADJ
ejpam-4412	600	15	or	or	CCONJ
ejpam-4412	600	16	log	log	NOUN
ejpam-4412	600	17	-	-	PUNCT
ejpam-4412	600	18	hyponormal	hyponormal	NOUN
ejpam-4412	600	19	.	.	PUNCT
ejpam-4412	601	1	let	let	VERB
ejpam-4412	601	2	s	s	PRON
ejpam-4412	601	3	∈	∈	PROPN
ejpam-4412	601	4	b(k	b(k	PROPN
ejpam-4412	601	5	)	)	PUNCT
ejpam-4412	601	6	be	be	AUX
ejpam-4412	601	7	class	class	NOUN
ejpam-4412	601	8	q	q	NOUN
ejpam-4412	601	9	-	-	PUNCT
ejpam-4412	601	10	wa(s	wa(s	NUM
ejpam-4412	601	11	,	,	PUNCT
ejpam-4412	601	12	t	t	PROPN
ejpam-4412	601	13	)	)	PUNCT
ejpam-4412	601	14	with	with	ADP
ejpam-4412	601	15	ker(s	ker(s	NOUN
ejpam-4412	601	16	)	)	PUNCT
ejpam-4412	601	17	⊂	⊂	X
ejpam-4412	601	18	ker(s∗	ker(s∗	PROPN
ejpam-4412	601	19	)	)	PUNCT
ejpam-4412	601	20	.	.	PUNCT
ejpam-4412	602	1	if	if	SCONJ
ejpam-4412	602	2	xt	xt	PROPN
ejpam-4412	602	3	=	=	SYM
ejpam-4412	602	4	sx	sx	PROPN
ejpam-4412	602	5	,	,	PUNCT
ejpam-4412	602	6	for	for	ADP
ejpam-4412	602	7	some	some	DET
ejpam-4412	602	8	x	x	SYM
ejpam-4412	602	9	∈	∈	PROPN
ejpam-4412	602	10	b(h	b(h	PROPN
ejpam-4412	602	11	,	,	PUNCT
ejpam-4412	602	12	k	k	NOUN
ejpam-4412	602	13	)	)	PUNCT
ejpam-4412	602	14	.	.	PUNCT
ejpam-4412	603	1	then	then	ADV
ejpam-4412	603	2	xt	xt	ADP
ejpam-4412	603	3	∗	∗	NOUN
ejpam-4412	603	4	=	=	SYM
ejpam-4412	603	5	s∗x	s∗x	NOUN
ejpam-4412	603	6	.	.	PUNCT
ejpam-4412	603	7	proof	proof	NOUN
ejpam-4412	603	8	.	.	PUNCT
ejpam-4412	604	1	let	let	AUX
ejpam-4412	604	2	t	t	PROPN
ejpam-4412	604	3	∗	∗	NOUN
ejpam-4412	604	4	be	be	AUX
ejpam-4412	604	5	a	a	DET
ejpam-4412	604	6	p	p	ADJ
ejpam-4412	604	7	-	-	PUNCT
ejpam-4412	604	8	hyponormal	hyponormal	ADJ
ejpam-4412	604	9	operator	operator	NOUN
ejpam-4412	604	10	for	for	ADP
ejpam-4412	604	11	p	p	PRON
ejpam-4412	604	12	≥	≥	NUM
ejpam-4412	604	13	1	1	NUM
ejpam-4412	604	14	2	2	NUM
ejpam-4412	604	15	and	and	CCONJ
ejpam-4412	604	16	let	let	VERB
ejpam-4412	604	17	t	t	NOUN
ejpam-4412	604	18	=	=	SYM
ejpam-4412	604	19	u	u	NOUN
ejpam-4412	604	20	|t	|t	VERB
ejpam-4412	604	21	|	|	ADV
ejpam-4412	604	22	be	be	AUX
ejpam-4412	604	23	the	the	DET
ejpam-4412	604	24	polar	polar	ADJ
ejpam-4412	604	25	decomposition	decomposition	NOUN
ejpam-4412	604	26	of	of	ADP
ejpam-4412	604	27	t	t	PROPN
ejpam-4412	604	28	.	.	PUNCT
ejpam-4412	605	1	then	then	ADV
ejpam-4412	605	2	the	the	DET
ejpam-4412	605	3	generalized	generalized	ADJ
ejpam-4412	605	4	aluthge	aluthge	ADJ
ejpam-4412	605	5	transform	transform	NOUN
ejpam-4412	605	6	t	t	PROPN
ejpam-4412	605	7	∗(s	∗(s	PROPN
ejpam-4412	605	8	,	,	PUNCT
ejpam-4412	605	9	t	t	PROPN
ejpam-4412	605	10	)	)	PUNCT
ejpam-4412	605	11	of	of	ADP
ejpam-4412	605	12	t	t	PROPN
ejpam-4412	605	13	∗	∗	NOUN
ejpam-4412	605	14	is	be	AUX
ejpam-4412	605	15	hyponormal	hyponormal	ADJ
ejpam-4412	605	16	and	and	CCONJ
ejpam-4412	605	17	satisfies	satisfie	NOUN
ejpam-4412	605	18	|t	|t	VERB
ejpam-4412	605	19	∗(s	∗(s	PROPN
ejpam-4412	605	20	,	,	PUNCT
ejpam-4412	605	21	t)|2	t)|2	PROPN
ejpam-4412	605	22	≥	≥	NUM
ejpam-4412	605	23	|t	|t	PROPN
ejpam-4412	605	24	|2	|2	NUM
ejpam-4412	605	25	≥	≥	NOUN
ejpam-4412	605	26	|(t	|(t	ADP
ejpam-4412	605	27	∗(s	∗(s	PROPN
ejpam-4412	605	28	,	,	PUNCT
ejpam-4412	605	29	t))∗|2	t))∗|2	NOUN
ejpam-4412	605	30	,	,	PUNCT
ejpam-4412	605	31	(	(	PUNCT
ejpam-4412	605	32	12	12	NUM
ejpam-4412	605	33	)	)	PUNCT
ejpam-4412	605	34	x	x	SYM
ejpam-4412	605	35	′t	′t	NOUN
ejpam-4412	605	36	(	(	PUNCT
ejpam-4412	605	37	s	s	PROPN
ejpam-4412	605	38	,	,	PUNCT
ejpam-4412	605	39	t	t	PROPN
ejpam-4412	605	40	)	)	PUNCT
ejpam-4412	606	1	=	=	SYM
ejpam-4412	606	2	sx	sx	INTJ
ejpam-4412	606	3	′	′	NUM
ejpam-4412	606	4	(	(	PUNCT
ejpam-4412	606	5	13	13	NUM
ejpam-4412	606	6	)	)	PUNCT
ejpam-4412	606	7	wherex	wherex	ADJ
ejpam-4412	606	8	′	′	NOUN
ejpam-4412	606	9	=	=	SYM
ejpam-4412	606	10	xu	xu	PROPN
ejpam-4412	606	11	|t	|t	PROPN
ejpam-4412	606	12	|t	|t	PROPN
ejpam-4412	606	13	.	.	PUNCT
ejpam-4412	607	1	using	use	VERB
ejpam-4412	607	2	the	the	DET
ejpam-4412	607	3	decompositionsh	decompositionsh	NOUN
ejpam-4412	607	4	=	=	SYM
ejpam-4412	607	5	ker(x	ker(x	PROPN
ejpam-4412	607	6	′)⊥⊕ker(x	′)⊥⊕ker(x	PROPN
ejpam-4412	607	7	′	′	NOUN
ejpam-4412	607	8	)	)	PUNCT
ejpam-4412	607	9	and	and	CCONJ
ejpam-4412	607	10	k	k	X
ejpam-4412	607	11	=	=	PUNCT
ejpam-4412	607	12	ran(x	ran(x	PROPN
ejpam-4412	607	13	′)⊕	′)⊕	X
ejpam-4412	607	14	ran(x	ran(x	ADP
ejpam-4412	607	15	′)⊥	′)⊥	NOUN
ejpam-4412	607	16	,	,	PUNCT
ejpam-4412	607	17	we	we	PRON
ejpam-4412	607	18	see	see	VERB
ejpam-4412	607	19	that	that	SCONJ
ejpam-4412	607	20	t	t	PROPN
ejpam-4412	607	21	(	(	PUNCT
ejpam-4412	607	22	s	s	PROPN
ejpam-4412	607	23	,	,	PUNCT
ejpam-4412	607	24	t	t	PROPN
ejpam-4412	607	25	)	)	PUNCT
ejpam-4412	607	26	,	,	PUNCT
ejpam-4412	607	27	s	s	VERB
ejpam-4412	607	28	and	and	CCONJ
ejpam-4412	607	29	x	x	SYM
ejpam-4412	607	30	′	′	NOUN
ejpam-4412	607	31	are	be	AUX
ejpam-4412	607	32	of	of	ADP
ejpam-4412	607	33	the	the	DET
ejpam-4412	607	34	form	form	NOUN
ejpam-4412	607	35	t	t	PROPN
ejpam-4412	607	36	∗(s	∗(s	PROPN
ejpam-4412	607	37	,	,	PUNCT
ejpam-4412	607	38	t	t	PROPN
ejpam-4412	607	39	)	)	PUNCT
ejpam-4412	607	40	=	=	PUNCT
ejpam-4412	608	1	(	(	PUNCT
ejpam-4412	608	2	t1	t1	NOUN
ejpam-4412	608	3	0	0	NUM
ejpam-4412	608	4	t2	t2	PROPN
ejpam-4412	608	5	t3	t3	PROPN
ejpam-4412	608	6	)	)	PUNCT
ejpam-4412	608	7	,	,	PUNCT
ejpam-4412	608	8	s	s	X
ejpam-4412	608	9	=	=	PUNCT
ejpam-4412	608	10	(	(	PUNCT
ejpam-4412	608	11	s1	s1	PROPN
ejpam-4412	608	12	s2	s2	PROPN
ejpam-4412	608	13	0	0	NUM
ejpam-4412	608	14	s3	s3	PROPN
ejpam-4412	608	15	)	)	PUNCT
ejpam-4412	608	16	,	,	PUNCT
ejpam-4412	608	17	x	x	X
ejpam-4412	608	18	′	′	NOUN
ejpam-4412	608	19	=	=	PUNCT
ejpam-4412	609	1	(	(	PUNCT
ejpam-4412	609	2	x1	x1	PROPN
ejpam-4412	609	3	0	0	NUM
ejpam-4412	609	4	0	0	NUM
ejpam-4412	609	5	0	0	NUM
ejpam-4412	609	6	)	)	PUNCT
ejpam-4412	609	7	where	where	SCONJ
ejpam-4412	609	8	t	t	PROPN
ejpam-4412	609	9	∗	∗	X
ejpam-4412	609	10	1	1	NUM
ejpam-4412	609	11	is	be	AUX
ejpam-4412	609	12	hyponormal	hyponormal	ADJ
ejpam-4412	609	13	,	,	PUNCT
ejpam-4412	609	14	s1	s1	PROPN
ejpam-4412	609	15	is	be	AUX
ejpam-4412	609	16	class	class	NOUN
ejpam-4412	609	17	q	q	NOUN
ejpam-4412	609	18	-	-	PUNCT
ejpam-4412	609	19	wa(s	wa(s	NUM
ejpam-4412	609	20	,	,	PUNCT
ejpam-4412	609	21	t	t	PROPN
ejpam-4412	609	22	)	)	PUNCT
ejpam-4412	609	23	with	with	ADP
ejpam-4412	609	24	ker(s1	ker(s1	PROPN
ejpam-4412	609	25	)	)	PUNCT
ejpam-4412	609	26	⊂	⊂	PRON
ejpam-4412	610	1	ker(s∗	ker(s∗	X
ejpam-4412	611	1	1	1	X
ejpam-4412	611	2	)	)	PUNCT
ejpam-4412	611	3	and	and	CCONJ
ejpam-4412	611	4	x1	x1	PROPN
ejpam-4412	611	5	is	be	AUX
ejpam-4412	611	6	a	a	DET
ejpam-4412	611	7	one	one	NUM
ejpam-4412	611	8	-	-	PUNCT
ejpam-4412	611	9	one	one	NUM
ejpam-4412	611	10	operator	operator	NOUN
ejpam-4412	611	11	with	with	ADP
ejpam-4412	611	12	dense	dense	ADJ
ejpam-4412	611	13	range	range	NOUN
ejpam-4412	611	14	.	.	PUNCT
ejpam-4412	612	1	since	since	SCONJ
ejpam-4412	612	2	x	x	SYM
ejpam-4412	612	3	′t	′t	X
ejpam-4412	612	4	(	(	PUNCT
ejpam-4412	612	5	s	s	PROPN
ejpam-4412	612	6	,	,	PUNCT
ejpam-4412	612	7	t	t	PROPN
ejpam-4412	612	8	)	)	PUNCT
ejpam-4412	612	9	=	=	SYM
ejpam-4412	612	10	sx	sx	PROPN
ejpam-4412	612	11	′	′	NOUN
ejpam-4412	612	12	,	,	PUNCT
ejpam-4412	612	13	we	we	PRON
ejpam-4412	612	14	have	have	VERB
ejpam-4412	612	15	x1t1	x1t1	NOUN
ejpam-4412	612	16	=	=	SYM
ejpam-4412	612	17	s1x1	s1x1	NOUN
ejpam-4412	612	18	.	.	PUNCT
ejpam-4412	613	1	(	(	PUNCT
ejpam-4412	613	2	14	14	NUM
ejpam-4412	613	3	)	)	PUNCT
ejpam-4412	613	4	hence	hence	ADV
ejpam-4412	613	5	t1	t1	NOUN
ejpam-4412	613	6	and	and	CCONJ
ejpam-4412	613	7	s1	s1	NOUN
ejpam-4412	613	8	are	be	AUX
ejpam-4412	613	9	normal	normal	ADJ
ejpam-4412	613	10	by	by	ADP
ejpam-4412	613	11	corollary	corollary	ADJ
ejpam-4412	613	12	6	6	NUM
ejpam-4412	613	13	,	,	PUNCT
ejpam-4412	613	14	so	so	SCONJ
ejpam-4412	613	15	that	that	SCONJ
ejpam-4412	613	16	t2	t2	NOUN
ejpam-4412	613	17	=	=	SYM
ejpam-4412	613	18	0	0	NUM
ejpam-4412	613	19	,	,	PUNCT
ejpam-4412	613	20	by	by	ADP
ejpam-4412	613	21	lemma	lemma	PROPN
ejpam-4412	613	22	12	12	NUM
ejpam-4412	613	23	of	of	ADP
ejpam-4412	613	24	[	[	X
ejpam-4412	613	25	30	30	NUM
ejpam-4412	613	26	]	]	PUNCT
ejpam-4412	613	27	and	and	CCONJ
ejpam-4412	613	28	s2	s2	X
ejpam-4412	613	29	=	=	SYM
ejpam-4412	613	30	0	0	NUM
ejpam-4412	613	31	by	by	ADP
ejpam-4412	613	32	lemma	lemma	PROPN
ejpam-4412	613	33	7	7	NUM
ejpam-4412	613	34	.	.	PUNCT
ejpam-4412	614	1	then	then	ADV
ejpam-4412	614	2	|t	|t	VERB
ejpam-4412	614	3	|	|	ADV
ejpam-4412	614	4	=	=	SYM
ejpam-4412	614	5	|t1|⊕p	|t1|⊕p	PROPN
ejpam-4412	614	6	,	,	PUNCT
ejpam-4412	614	7	for	for	ADP
ejpam-4412	614	8	some	some	DET
ejpam-4412	614	9	positive	positive	ADJ
ejpam-4412	614	10	operator	operator	NOUN
ejpam-4412	614	11	p	p	NOUN
ejpam-4412	614	12	,	,	PUNCT
ejpam-4412	614	13	by	by	ADP
ejpam-4412	614	14	(	(	PUNCT
ejpam-4412	614	15	12	12	NUM
ejpam-4412	614	16	)	)	PUNCT
ejpam-4412	614	17	and	and	CCONJ
ejpam-4412	614	18	u	u	X
ejpam-4412	614	19	=	=	PUNCT
ejpam-4412	614	20	(	(	PUNCT
ejpam-4412	614	21	u1	u1	NOUN
ejpam-4412	614	22	u2	u2	PROPN
ejpam-4412	614	23	0	0	NUM
ejpam-4412	614	24	u3	u3	PROPN
ejpam-4412	614	25	)	)	PUNCT
ejpam-4412	614	26	by	by	ADP
ejpam-4412	614	27	lemma	lemma	PROPN
ejpam-4412	614	28	13	13	NUM
ejpam-4412	614	29	of	of	ADP
ejpam-4412	614	30	[	[	X
ejpam-4412	614	31	30	30	NUM
ejpam-4412	614	32	]	]	PUNCT
ejpam-4412	614	33	.	.	PUNCT
ejpam-4412	615	1	let	let	VERB
ejpam-4412	615	2	x	x	PUNCT
ejpam-4412	615	3	=	=	PRON
ejpam-4412	615	4	(	(	PUNCT
ejpam-4412	615	5	x11	x11	NOUN
ejpam-4412	615	6	x12	x12	NUM
ejpam-4412	615	7	x21	x21	PROPN
ejpam-4412	615	8	x22	x22	PROPN
ejpam-4412	615	9	)	)	PUNCT
ejpam-4412	615	10	be	be	AUX
ejpam-4412	615	11	a	a	DET
ejpam-4412	615	12	2	2	NUM
ejpam-4412	615	13	×	×	NOUN
ejpam-4412	615	14	2	2	NUM
ejpam-4412	615	15	matrix	matrix	NOUN
ejpam-4412	615	16	representation	representation	NOUN
ejpam-4412	615	17	of	of	ADP
ejpam-4412	615	18	x	x	PUNCT
ejpam-4412	615	19	with	with	ADP
ejpam-4412	615	20	respect	respect	NOUN
ejpam-4412	615	21	to	to	ADP
ejpam-4412	615	22	the	the	DET
ejpam-4412	615	23	decomposition	decomposition	NOUN
ejpam-4412	615	24	h	h	NOUN
ejpam-4412	615	25	=	=	SYM
ejpam-4412	615	26	ker(x	ker(x	PROPN
ejpam-4412	615	27	′)⊥⊕ker(x	′)⊥⊕ker(x	PROPN
ejpam-4412	615	28	′	′	NOUN
ejpam-4412	615	29	)	)	PUNCT
ejpam-4412	615	30	and	and	CCONJ
ejpam-4412	616	1	k	k	X
ejpam-4412	616	2	=	=	PUNCT
ejpam-4412	616	3	ran(x	ran(x	PROPN
ejpam-4412	616	4	′)⊕	′)⊕	NOUN
ejpam-4412	616	5	ran(x	ran(x	ADP
ejpam-4412	616	6	′)⊥.	′)⊥.	NOUN
ejpam-4412	616	7	then	then	ADV
ejpam-4412	616	8	x	x	ADP
ejpam-4412	616	9	′	′	NOUN
ejpam-4412	616	10	=	=	SYM
ejpam-4412	616	11	xu	xu	PROPN
ejpam-4412	616	12	|t	|t	PROPN
ejpam-4412	616	13	|t	|t	PROPN
ejpam-4412	616	14	implies	imply	VERB
ejpam-4412	616	15	that	that	SCONJ
ejpam-4412	616	16	x1	x1	PROPN
ejpam-4412	616	17	=	=	PUNCT
ejpam-4412	617	1	x11u1|t1|t	x11u1|t1|t	PROPN
ejpam-4412	617	2	and	and	CCONJ
ejpam-4412	617	3	hence	hence	ADV
ejpam-4412	617	4	ker(t1	ker(t1	PROPN
ejpam-4412	617	5	)	)	PUNCT
ejpam-4412	618	1	⊂	⊂	PROPN
ejpam-4412	619	1	ker(x1	ker(x1	PROPN
ejpam-4412	619	2	)	)	PUNCT
ejpam-4412	620	1	=	=	PRON
ejpam-4412	620	2	{	{	PUNCT
ejpam-4412	620	3	0	0	NUM
ejpam-4412	620	4	}	}	PUNCT
ejpam-4412	620	5	.	.	PUNCT
ejpam-4412	621	1	this	this	PRON
ejpam-4412	621	2	shows	show	VERB
ejpam-4412	621	3	that	that	SCONJ
ejpam-4412	621	4	t1	t1	NOUN
ejpam-4412	621	5	is	be	AUX
ejpam-4412	621	6	one	one	NUM
ejpam-4412	621	7	-	-	PUNCT
ejpam-4412	621	8	one	one	NUM
ejpam-4412	621	9	and	and	CCONJ
ejpam-4412	621	10	hence	hence	ADV
ejpam-4412	621	11	it	it	PRON
ejpam-4412	621	12	has	have	VERB
ejpam-4412	621	13	dense	dense	ADJ
ejpam-4412	621	14	range	range	NOUN
ejpam-4412	621	15	,	,	PUNCT
ejpam-4412	621	16	so	so	SCONJ
ejpam-4412	621	17	that	that	SCONJ
ejpam-4412	621	18	u2	u2	NOUN
ejpam-4412	621	19	=	=	NOUN
ejpam-4412	621	20	0	0	PROPN
ejpam-4412	621	21	and	and	CCONJ
ejpam-4412	621	22	t	t	PROPN
ejpam-4412	621	23	=	=	SYM
ejpam-4412	621	24	t1	t1	PROPN
ejpam-4412	621	25	⊕	⊕	PROPN
ejpam-4412	621	26	t4	t4	PROPN
ejpam-4412	621	27	for	for	ADP
ejpam-4412	621	28	some	some	DET
ejpam-4412	621	29	hyponormal	hyponormal	ADJ
ejpam-4412	621	30	operator	operator	NOUN
ejpam-4412	621	31	t	t	PROPN
ejpam-4412	621	32	∗	∗	X
ejpam-4412	621	33	4	4	NUM
ejpam-4412	621	34	by	by	ADP
ejpam-4412	621	35	[	[	X
ejpam-4412	621	36	30	30	NUM
ejpam-4412	621	37	,	,	PUNCT
ejpam-4412	621	38	lemma	lemma	PROPN
ejpam-4412	621	39	13	13	NUM
ejpam-4412	621	40	]	]	PUNCT
ejpam-4412	621	41	.	.	PUNCT
ejpam-4412	622	1	since	since	SCONJ
ejpam-4412	622	2	(	(	PUNCT
ejpam-4412	622	3	x1	x1	PROPN
ejpam-4412	622	4	0	0	NUM
ejpam-4412	622	5	0	0	NUM
ejpam-4412	622	6	0	0	NUM
ejpam-4412	622	7	)	)	PUNCT
ejpam-4412	622	8	=	=	PUNCT
ejpam-4412	623	1	x	x	PUNCT
ejpam-4412	623	2	′	′	NUM
ejpam-4412	623	3	=	=	SYM
ejpam-4412	623	4	xu	xu	PROPN
ejpam-4412	623	5	|t	|t	INTJ
ejpam-4412	623	6	|t	|t	PROPN
ejpam-4412	624	1	=	=	PUNCT
ejpam-4412	624	2	(	(	PUNCT
ejpam-4412	624	3	x11	x11	NUM
ejpam-4412	624	4	x12	x12	NUM
ejpam-4412	624	5	x21	x21	NUM
ejpam-4412	624	6	x22	x22	PROPN
ejpam-4412	624	7	)	)	PUNCT
ejpam-4412	624	8	(	(	PUNCT
ejpam-4412	624	9	u1|t1|t	u1|t1|t	PROPN
ejpam-4412	624	10	0	0	NUM
ejpam-4412	624	11	0	0	NUM
ejpam-4412	624	12	u3|t4|t	u3|t4|t	NOUN
ejpam-4412	624	13	)	)	PUNCT
ejpam-4412	624	14	we	we	PRON
ejpam-4412	624	15	deduce	deduce	VERB
ejpam-4412	624	16	the	the	DET
ejpam-4412	624	17	following	follow	VERB
ejpam-4412	624	18	assertions	assertion	NOUN
ejpam-4412	624	19	.	.	PUNCT
ejpam-4412	625	1	x12u2|t4|t	x12u2|t4|t	NOUN
ejpam-4412	625	2	=	=	PUNCT
ejpam-4412	625	3	0	0	NUM
ejpam-4412	625	4	;	;	PUNCT
ejpam-4412	625	5	hence	hence	ADV
ejpam-4412	625	6	x12t3	x12t3	PUNCT
ejpam-4412	626	1	=	=	PUNCT
ejpam-4412	626	2	0	0	PUNCT
ejpam-4412	626	3	because	because	SCONJ
ejpam-4412	626	4	t4	t4	PROPN
ejpam-4412	626	5	=	=	PROPN
ejpam-4412	626	6	u3|t4|	u3|t4|	PROPN
ejpam-4412	626	7	.	.	PUNCT
ejpam-4412	627	1	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	627	2	,	,	PUNCT
ejpam-4412	627	3	n.	n.	NOUN
ejpam-4412	627	4	h.	h.	PROPN
ejpam-4412	627	5	altaweel	altaweel	PROPN
ejpam-4412	627	6	/	/	SYM
ejpam-4412	627	7	eur	eur	PROPN
ejpam-4412	627	8	.	.	PUNCT
ejpam-4412	628	1	j.	j.	PROPN
ejpam-4412	628	2	pure	pure	PROPN
ejpam-4412	628	3	appl	appl	PROPN
ejpam-4412	628	4	.	.	PROPN
ejpam-4412	628	5	math	math	PROPN
ejpam-4412	628	6	,	,	PUNCT
ejpam-4412	628	7	15	15	NUM
ejpam-4412	628	8	(	(	PUNCT
ejpam-4412	628	9	3	3	NUM
ejpam-4412	628	10	)	)	PUNCT
ejpam-4412	628	11	(	(	PUNCT
ejpam-4412	628	12	2022	2022	NUM
ejpam-4412	628	13	)	)	PUNCT
ejpam-4412	628	14	,	,	PUNCT
ejpam-4412	628	15	1067	1067	NUM
ejpam-4412	628	16	-	-	SYM
ejpam-4412	628	17	1089	1089	NUM
ejpam-4412	628	18	1086	1086	NUM
ejpam-4412	628	19	x21u1|t1|t	x21u1|t1|t	NUM
ejpam-4412	628	20	;	;	PUNCT
ejpam-4412	628	21	hence	hence	ADV
ejpam-4412	628	22	x12	x12	NUM
ejpam-4412	628	23	=	=	SYM
ejpam-4412	628	24	0	0	NUM
ejpam-4412	628	25	because	because	SCONJ
ejpam-4412	628	26	u1|t1|	u1|t1|	PROPN
ejpam-4412	628	27	1	1	NUM
ejpam-4412	628	28	2	2	NUM
ejpam-4412	628	29	has	have	VERB
ejpam-4412	628	30	dense	dense	ADJ
ejpam-4412	628	31	range	range	NOUN
ejpam-4412	628	32	.	.	PUNCT
ejpam-4412	629	1	x22u3|t4|t	x22u3|t4|t	NOUN
ejpam-4412	629	2	=	=	PUNCT
ejpam-4412	629	3	0	0	NUM
ejpam-4412	629	4	;	;	PUNCT
ejpam-4412	629	5	hence	hence	ADV
ejpam-4412	629	6	x22t3	x22t3	PUNCT
ejpam-4412	630	1	=	=	PUNCT
ejpam-4412	630	2	0	0	X
ejpam-4412	630	3	.	.	PUNCT
ejpam-4412	631	1	the	the	DET
ejpam-4412	631	2	assumption	assumption	NOUN
ejpam-4412	631	3	xt	xt	PUNCT
ejpam-4412	632	1	=	=	SYM
ejpam-4412	632	2	sx	sx	PROPN
ejpam-4412	632	3	tell	tell	VERB
ejpam-4412	632	4	us	we	PRON
ejpam-4412	632	5	that	that	SCONJ
ejpam-4412	632	6	,	,	PUNCT
ejpam-4412	632	7	x11t1	x11t1	PROPN
ejpam-4412	632	8	=	=	X
ejpam-4412	632	9	s1x11	s1x11	NOUN
ejpam-4412	632	10	x12t4	x12t4	PUNCT
ejpam-4412	633	1	=	=	PUNCT
ejpam-4412	633	2	s1x12	s1x12	X
ejpam-4412	633	3	=	=	SYM
ejpam-4412	633	4	0	0	NUM
ejpam-4412	633	5	,	,	PUNCT
ejpam-4412	633	6	x22t4	x22t4	X
ejpam-4412	633	7	=	=	SYM
ejpam-4412	634	1	s3x22	s3x22	NOUN
ejpam-4412	634	2	=	=	NOUN
ejpam-4412	634	3	0	0	NUM
ejpam-4412	634	4	.	.	PUNCT
ejpam-4412	635	1	since	since	SCONJ
ejpam-4412	635	2	t1	t1	NOUN
ejpam-4412	635	3	and	and	CCONJ
ejpam-4412	635	4	s1	s1	NOUN
ejpam-4412	635	5	are	be	AUX
ejpam-4412	635	6	normal	normal	ADJ
ejpam-4412	635	7	,	,	PUNCT
ejpam-4412	635	8	we	we	PRON
ejpam-4412	635	9	have	have	VERB
ejpam-4412	635	10	x11	x11	PROPN
ejpam-4412	635	11	t	t	PROPN
ejpam-4412	635	12	∗	∗	X
ejpam-4412	635	13	1	1	NUM
ejpam-4412	635	14	=	=	NUM
ejpam-4412	635	15	s∗	s∗	PROPN
ejpam-4412	635	16	1x11	1x11	NUM
ejpam-4412	635	17	,	,	PUNCT
ejpam-4412	635	18	by	by	ADP
ejpam-4412	635	19	fuglede	fuglede	ADJ
ejpam-4412	635	20	-	-	PUNCT
ejpam-4412	635	21	putnam	putnam	NOUN
ejpam-4412	635	22	theorem	theorem	NOUN
ejpam-4412	635	23	.	.	PUNCT
ejpam-4412	636	1	the	the	DET
ejpam-4412	636	2	p	p	NOUN
ejpam-4412	636	3	-	-	PUNCT
ejpam-4412	636	4	hyponormality	hyponormality	NOUN
ejpam-4412	636	5	of	of	ADP
ejpam-4412	636	6	t	t	PROPN
ejpam-4412	636	7	∗	∗	X
ejpam-4412	636	8	4	4	NUM
ejpam-4412	636	9	shows	show	VERB
ejpam-4412	636	10	that	that	SCONJ
ejpam-4412	636	11	ran(t	ran(t	ADJ
ejpam-4412	636	12	∗	∗	NOUN
ejpam-4412	636	13	4	4	NUM
ejpam-4412	636	14	)	)	PUNCT
ejpam-4412	636	15	⊂	⊂	PROPN
ejpam-4412	636	16	ran(t4	ran(t4	PROPN
ejpam-4412	636	17	)	)	PUNCT
ejpam-4412	636	18	.	.	PUNCT
ejpam-4412	637	1	also	also	ADV
ejpam-4412	637	2	,	,	PUNCT
ejpam-4412	637	3	we	we	PRON
ejpam-4412	637	4	have	have	VERB
ejpam-4412	637	5	ker(s3	ker(s3	X
ejpam-4412	637	6	)	)	PUNCT
ejpam-4412	637	7	⊂	⊂	PROPN
ejpam-4412	637	8	ker(s∗	ker(s∗	X
ejpam-4412	637	9	3	3	NUM
ejpam-4412	637	10	)	)	PUNCT
ejpam-4412	637	11	.	.	PUNCT
ejpam-4412	638	1	hence	hence	ADV
ejpam-4412	638	2	,	,	PUNCT
ejpam-4412	638	3	we	we	PRON
ejpam-4412	638	4	also	also	ADV
ejpam-4412	638	5	have	have	VERB
ejpam-4412	638	6	x12	x12	NUM
ejpam-4412	638	7	t	t	PROPN
ejpam-4412	638	8	∗	∗	NOUN
ejpam-4412	638	9	4	4	NUM
ejpam-4412	638	10	=	=	SYM
ejpam-4412	638	11	s∗	s∗	PROPN
ejpam-4412	638	12	1x12	1x12	NUM
ejpam-4412	638	13	=	=	SYM
ejpam-4412	638	14	0	0	NUM
ejpam-4412	639	1	and	and	CCONJ
ejpam-4412	639	2	x22	x22	NUM
ejpam-4412	639	3	t	t	PROPN
ejpam-4412	639	4	∗	∗	NOUN
ejpam-4412	639	5	4	4	NUM
ejpam-4412	639	6	s	s	NOUN
ejpam-4412	639	7	∗	∗	NOUN
ejpam-4412	639	8	3x22	3x22	NUM
ejpam-4412	639	9	=	=	SYM
ejpam-4412	639	10	0	0	X
ejpam-4412	639	11	.	.	PUNCT
ejpam-4412	640	1	this	this	PRON
ejpam-4412	640	2	implies	imply	VERB
ejpam-4412	640	3	that	that	SCONJ
ejpam-4412	640	4	xt	xt	ADP
ejpam-4412	640	5	∗	∗	NOUN
ejpam-4412	640	6	=	=	SYM
ejpam-4412	640	7	x11	x11	PROPN
ejpam-4412	640	8	t	t	PROPN
ejpam-4412	640	9	∗	∗	X
ejpam-4412	640	10	1	1	NUM
ejpam-4412	640	11	⊕	⊕	NUM
ejpam-4412	640	12	0	0	NUM
ejpam-4412	641	1	=	=	PUNCT
ejpam-4412	641	2	s∗	s∗	PROPN
ejpam-4412	641	3	1x11	1x11	NUM
ejpam-4412	641	4	⊕	⊕	PROPN
ejpam-4412	641	5	0	0	NUM
ejpam-4412	642	1	=	=	SYM
ejpam-4412	642	2	s∗x	s∗x	PROPN
ejpam-4412	642	3	.	.	PUNCT
ejpam-4412	643	1	next	next	ADV
ejpam-4412	643	2	,	,	PUNCT
ejpam-4412	643	3	we	we	PRON
ejpam-4412	643	4	prove	prove	VERB
ejpam-4412	643	5	the	the	DET
ejpam-4412	643	6	case	case	NOUN
ejpam-4412	643	7	where	where	SCONJ
ejpam-4412	643	8	t	t	PROPN
ejpam-4412	643	9	∗	∗	NOUN
ejpam-4412	643	10	is	be	AUX
ejpam-4412	643	11	p	p	NOUN
ejpam-4412	643	12	-	-	PUNCT
ejpam-4412	643	13	hyponormal	hyponormal	NOUN
ejpam-4412	643	14	for	for	ADP
ejpam-4412	643	15	0	0	NUM
ejpam-4412	643	16	<	<	X
ejpam-4412	643	17	p	p	X
ejpam-4412	643	18	≤	≤	NUM
ejpam-4412	643	19	1	1	NUM
ejpam-4412	643	20	2	2	NUM
ejpam-4412	643	21	.	.	PUNCT
ejpam-4412	644	1	let	let	VERB
ejpam-4412	644	2	x	x	PRON
ejpam-4412	644	3	′	′	NUM
ejpam-4412	644	4	be	be	AUX
ejpam-4412	644	5	as	as	ADV
ejpam-4412	644	6	above	above	ADV
ejpam-4412	644	7	.	.	PUNCT
ejpam-4412	645	1	then	then	ADV
ejpam-4412	645	2	t	t	PROPN
ejpam-4412	645	3	∗(s	∗(s	PROPN
ejpam-4412	645	4	,	,	PUNCT
ejpam-4412	645	5	t	t	PROPN
ejpam-4412	645	6	)	)	PUNCT
ejpam-4412	645	7	is	be	AUX
ejpam-4412	645	8	(	(	PUNCT
ejpam-4412	645	9	p+	p+	NOUN
ejpam-4412	645	10	1	1	NUM
ejpam-4412	645	11	2)-hyponormal	2)-hyponormal	NUM
ejpam-4412	645	12	and	and	CCONJ
ejpam-4412	645	13	satisfies	satisfy	VERB
ejpam-4412	645	14	x	x	X
ejpam-4412	645	15	′t	′t	X
ejpam-4412	645	16	(	(	PUNCT
ejpam-4412	645	17	s	s	PROPN
ejpam-4412	645	18	,	,	PUNCT
ejpam-4412	645	19	t	t	PROPN
ejpam-4412	645	20	)	)	PUNCT
ejpam-4412	646	1	=	=	SYM
ejpam-4412	646	2	sx	sx	PROPN
ejpam-4412	646	3	′.	′.	NOUN
ejpam-4412	646	4	use	use	VERB
ejpam-4412	646	5	the	the	DET
ejpam-4412	646	6	same	same	ADJ
ejpam-4412	646	7	argument	argument	NOUN
ejpam-4412	646	8	as	as	ADP
ejpam-4412	646	9	above	above	ADV
ejpam-4412	646	10	.	.	PUNCT
ejpam-4412	647	1	we	we	PRON
ejpam-4412	647	2	obtain	obtain	VERB
ejpam-4412	647	3	t	t	PROPN
ejpam-4412	647	4	(	(	PUNCT
ejpam-4412	647	5	s	s	PROPN
ejpam-4412	647	6	,	,	PUNCT
ejpam-4412	647	7	t	t	PROPN
ejpam-4412	647	8	)	)	PUNCT
ejpam-4412	647	9	=	=	SYM
ejpam-4412	647	10	t1⊕t3	t1⊕t3	PROPN
ejpam-4412	647	11	on	on	ADP
ejpam-4412	647	12	h	h	PROPN
ejpam-4412	647	13	=	=	SYM
ejpam-4412	647	14	ker(x	ker(x	PROPN
ejpam-4412	647	15	′)⊥⊕ker(x	′)⊥⊕ker(x	PROPN
ejpam-4412	647	16	′	′	NOUN
ejpam-4412	647	17	)	)	PUNCT
ejpam-4412	647	18	and	and	CCONJ
ejpam-4412	647	19	s	s	NOUN
ejpam-4412	647	20	=	=	PUNCT
ejpam-4412	647	21	s1⊕s3	s1⊕s3	NOUN
ejpam-4412	647	22	,	,	PUNCT
ejpam-4412	647	23	where	where	SCONJ
ejpam-4412	647	24	t1	t1	NOUN
ejpam-4412	647	25	is	be	AUX
ejpam-4412	647	26	an	an	DET
ejpam-4412	647	27	injective	injective	ADJ
ejpam-4412	647	28	normal	normal	ADJ
ejpam-4412	647	29	operator	operator	NOUN
ejpam-4412	647	30	and	and	CCONJ
ejpam-4412	647	31	s1	s1	NOUN
ejpam-4412	647	32	is	be	AUX
ejpam-4412	647	33	also	also	ADV
ejpam-4412	647	34	normal	normal	ADJ
ejpam-4412	647	35	.	.	PUNCT
ejpam-4412	648	1	hence	hence	ADV
ejpam-4412	648	2	we	we	PRON
ejpam-4412	648	3	have	have	VERB
ejpam-4412	648	4	t	t	PROPN
ejpam-4412	648	5	=	=	SYM
ejpam-4412	648	6	t1	t1	PROPN
ejpam-4412	648	7	⊕	⊕	PROPN
ejpam-4412	648	8	t4	t4	PROPN
ejpam-4412	648	9	for	for	ADP
ejpam-4412	648	10	some	some	DET
ejpam-4412	648	11	p	p	PROPN
ejpam-4412	648	12	-	-	PUNCT
ejpam-4412	648	13	hyponormal	hyponormal	ADJ
ejpam-4412	648	14	t	t	PROPN
ejpam-4412	648	15	∗	∗	X
ejpam-4412	648	16	4	4	NUM
ejpam-4412	648	17	,	,	PUNCT
ejpam-4412	648	18	by	by	ADP
ejpam-4412	648	19	lemma	lemma	PROPN
ejpam-4412	648	20	13	13	NUM
ejpam-4412	648	21	of	of	ADP
ejpam-4412	648	22	[	[	X
ejpam-4412	648	23	30	30	NUM
ejpam-4412	648	24	]	]	PUNCT
ejpam-4412	648	25	.	.	PUNCT
ejpam-4412	649	1	again	again	ADV
ejpam-4412	649	2	using	use	VERB
ejpam-4412	649	3	the	the	DET
ejpam-4412	649	4	same	same	ADJ
ejpam-4412	649	5	argument	argument	NOUN
ejpam-4412	649	6	as	as	ADP
ejpam-4412	649	7	above	above	ADV
ejpam-4412	649	8	,	,	PUNCT
ejpam-4412	649	9	we	we	PRON
ejpam-4412	649	10	obtain	obtain	VERB
ejpam-4412	649	11	x21	x21	NUM
ejpam-4412	649	12	=	=	SYM
ejpam-4412	649	13	0	0	NUM
ejpam-4412	649	14	,	,	PUNCT
ejpam-4412	649	15	x11	x11	PROPN
ejpam-4412	649	16	t	t	PROPN
ejpam-4412	649	17	∗	∗	NOUN
ejpam-4412	649	18	1	1	NUM
ejpam-4412	649	19	=	=	NUM
ejpam-4412	649	20	s∗	s∗	PROPN
ejpam-4412	649	21	1x11	1x11	NUM
ejpam-4412	649	22	,	,	PUNCT
ejpam-4412	649	23	x12	x12	PROPN
ejpam-4412	649	24	t	t	PROPN
ejpam-4412	649	25	∗	∗	NOUN
ejpam-4412	649	26	4	4	NUM
ejpam-4412	649	27	=	=	SYM
ejpam-4412	649	28	s∗	s∗	PROPN
ejpam-4412	649	29	1x12	1x12	NUM
ejpam-4412	649	30	=	=	SYM
ejpam-4412	649	31	0	0	NUM
ejpam-4412	649	32	and	and	CCONJ
ejpam-4412	649	33	x22	x22	NUM
ejpam-4412	649	34	t	t	PROPN
ejpam-4412	649	35	∗	∗	NOUN
ejpam-4412	649	36	4	4	NUM
ejpam-4412	649	37	=	=	NUM
ejpam-4412	649	38	s∗	s∗	PROPN
ejpam-4412	649	39	3x22	3x22	NUM
ejpam-4412	649	40	=	=	SYM
ejpam-4412	649	41	0	0	X
ejpam-4412	649	42	.	.	PUNCT
ejpam-4412	650	1	hence	hence	ADV
ejpam-4412	650	2	we	we	PRON
ejpam-4412	650	3	have	have	VERB
ejpam-4412	650	4	xt	xt	X
ejpam-4412	650	5	∗	∗	NOUN
ejpam-4412	650	6	=	=	SYM
ejpam-4412	650	7	s∗x	s∗x	PROPN
ejpam-4412	650	8	.	.	PUNCT
ejpam-4412	651	1	finally	finally	ADV
ejpam-4412	651	2	,	,	PUNCT
ejpam-4412	651	3	we	we	PRON
ejpam-4412	651	4	assume	assume	VERB
ejpam-4412	651	5	that	that	SCONJ
ejpam-4412	651	6	t	t	PROPN
ejpam-4412	651	7	∗	∗	NOUN
ejpam-4412	651	8	is	be	AUX
ejpam-4412	651	9	log	log	NOUN
ejpam-4412	651	10	-	-	PUNCT
ejpam-4412	651	11	hyponormal	hyponormal	NOUN
ejpam-4412	651	12	.	.	PUNCT
ejpam-4412	652	1	let	let	VERB
ejpam-4412	652	2	t	t	PROPN
ejpam-4412	652	3	(	(	PUNCT
ejpam-4412	652	4	s	s	PROPN
ejpam-4412	652	5	,	,	PUNCT
ejpam-4412	652	6	t	t	PROPN
ejpam-4412	652	7	)	)	PUNCT
ejpam-4412	652	8	and	and	CCONJ
ejpam-4412	652	9	x	x	X
ejpam-4412	652	10	′	′	NUM
ejpam-4412	652	11	be	be	VERB
ejpam-4412	652	12	as	as	ADV
ejpam-4412	652	13	above	above	ADV
ejpam-4412	652	14	.	.	PUNCT
ejpam-4412	653	1	then	then	ADV
ejpam-4412	653	2	x	x	SYM
ejpam-4412	653	3	′t	′t	NOUN
ejpam-4412	653	4	(	(	PUNCT
ejpam-4412	653	5	s	s	PROPN
ejpam-4412	653	6	,	,	PUNCT
ejpam-4412	653	7	t	t	PROPN
ejpam-4412	653	8	)	)	PUNCT
ejpam-4412	653	9	=	=	VERB
ejpam-4412	653	10	sx	sx	INTJ
ejpam-4412	653	11	′	′	NOUN
ejpam-4412	654	1	and	and	CCONJ
ejpam-4412	654	2	t	t	PROPN
ejpam-4412	654	3	∗(s	∗(s	PROPN
ejpam-4412	654	4	,	,	PUNCT
ejpam-4412	654	5	t	t	PROPN
ejpam-4412	654	6	)	)	PUNCT
ejpam-4412	654	7	is	be	AUX
ejpam-4412	654	8	semi	semi	ADJ
ejpam-4412	654	9	-	-	ADJ
ejpam-4412	654	10	hyponormal	hyponormal	ADJ
ejpam-4412	654	11	and	and	CCONJ
ejpam-4412	654	12	satisfies	satisfie	NOUN
ejpam-4412	654	13	|t	|t	VERB
ejpam-4412	654	14	∗(s	∗(s	PROPN
ejpam-4412	654	15	,	,	PUNCT
ejpam-4412	654	16	t)|	t)|	ADJ
ejpam-4412	654	17	≥	≥	NOUN
ejpam-4412	654	18	|t	|t	VERB
ejpam-4412	655	1	∗|	∗|	PROPN
ejpam-4412	655	2	≥	≥	NUM
ejpam-4412	655	3	|(t	|(t	PROPN
ejpam-4412	655	4	∗(s	∗(s	PROPN
ejpam-4412	655	5	,	,	PUNCT
ejpam-4412	655	6	t)∗|	t)∗|	NUM
ejpam-4412	655	7	.	.	PUNCT
ejpam-4412	656	1	by	by	ADP
ejpam-4412	656	2	the	the	DET
ejpam-4412	656	3	same	same	ADJ
ejpam-4412	656	4	argument	argument	NOUN
ejpam-4412	656	5	as	as	ADP
ejpam-4412	656	6	above	above	ADV
ejpam-4412	656	7	,	,	PUNCT
ejpam-4412	656	8	we	we	PRON
ejpam-4412	656	9	have	have	VERB
ejpam-4412	656	10	t	t	PROPN
ejpam-4412	656	11	(	(	PUNCT
ejpam-4412	656	12	s	s	PROPN
ejpam-4412	656	13	,	,	PUNCT
ejpam-4412	656	14	t	t	PROPN
ejpam-4412	656	15	)	)	PUNCT
ejpam-4412	656	16	=	=	SYM
ejpam-4412	657	1	t1⊕t3	t1⊕t3	PROPN
ejpam-4412	657	2	on	on	ADP
ejpam-4412	657	3	h	h	PROPN
ejpam-4412	657	4	=	=	SYM
ejpam-4412	657	5	ker(x	ker(x	PROPN
ejpam-4412	657	6	′)⊥⊕ker(x	′)⊥⊕ker(x	PROPN
ejpam-4412	657	7	′	′	NOUN
ejpam-4412	657	8	)	)	PUNCT
ejpam-4412	657	9	and	and	CCONJ
ejpam-4412	657	10	s	s	NOUN
ejpam-4412	657	11	=	=	SYM
ejpam-4412	657	12	s1	s1	PROPN
ejpam-4412	657	13	⊕	⊕	PROPN
ejpam-4412	657	14	s3	s3	PROPN
ejpam-4412	657	15	on	on	ADP
ejpam-4412	657	16	k	k	PROPN
ejpam-4412	657	17	=	=	PUNCT
ejpam-4412	657	18	ran(x	ran(x	PROPN
ejpam-4412	657	19	′)⊕	′)⊕	X
ejpam-4412	657	20	ran(x	ran(x	ADP
ejpam-4412	657	21	′)⊥	′)⊥	NOUN
ejpam-4412	657	22	,	,	PUNCT
ejpam-4412	657	23	where	where	SCONJ
ejpam-4412	657	24	t1	t1	NOUN
ejpam-4412	657	25	is	be	AUX
ejpam-4412	657	26	an	an	DET
ejpam-4412	657	27	injective	injective	ADJ
ejpam-4412	657	28	normal	normal	ADJ
ejpam-4412	657	29	operator	operator	NOUN
ejpam-4412	657	30	,	,	PUNCT
ejpam-4412	657	31	s1	s1	PROPN
ejpam-4412	657	32	is	be	AUX
ejpam-4412	657	33	normal	normal	ADJ
ejpam-4412	657	34	,	,	PUNCT
ejpam-4412	657	35	t	t	PROPN
ejpam-4412	657	36	∗	∗	NOUN
ejpam-4412	657	37	3	3	NUM
ejpam-4412	657	38	is	be	AUX
ejpam-4412	657	39	invertible	invertible	ADJ
ejpam-4412	657	40	semi	semi	ADJ
ejpam-4412	657	41	-	-	ADJ
ejpam-4412	657	42	hyponormal	hyponormal	ADJ
ejpam-4412	657	43	and	and	CCONJ
ejpam-4412	657	44	s3	s3	PROPN
ejpam-4412	657	45	is	be	AUX
ejpam-4412	657	46	class	class	NOUN
ejpam-4412	657	47	q	q	NOUN
ejpam-4412	657	48	-	-	PUNCT
ejpam-4412	657	49	wa(s	wa(s	NUM
ejpam-4412	657	50	,	,	PUNCT
ejpam-4412	657	51	t	t	PROPN
ejpam-4412	657	52	)	)	PUNCT
ejpam-4412	657	53	with	with	ADP
ejpam-4412	657	54	ker(s3	ker(s3	PROPN
ejpam-4412	657	55	)	)	PUNCT
ejpam-4412	657	56	⊂	⊂	PROPN
ejpam-4412	657	57	ker(s∗	ker(s∗	X
ejpam-4412	657	58	3	3	NUM
ejpam-4412	657	59	)	)	PUNCT
ejpam-4412	657	60	.	.	PUNCT
ejpam-4412	658	1	by	by	ADP
ejpam-4412	658	2	lemma	lemma	PROPN
ejpam-4412	658	3	13	13	NUM
ejpam-4412	658	4	of	of	ADP
ejpam-4412	658	5	[	[	X
ejpam-4412	658	6	30	30	NUM
ejpam-4412	658	7	]	]	PUNCT
ejpam-4412	658	8	,	,	PUNCT
ejpam-4412	658	9	we	we	PRON
ejpam-4412	658	10	have	have	VERB
ejpam-4412	658	11	that	that	DET
ejpam-4412	658	12	t	t	PROPN
ejpam-4412	658	13	is	be	AUX
ejpam-4412	658	14	of	of	ADP
ejpam-4412	658	15	the	the	DET
ejpam-4412	658	16	form	form	NOUN
ejpam-4412	658	17	t	t	PROPN
ejpam-4412	658	18	=	=	SYM
ejpam-4412	658	19	t1⊕t4	t1⊕t4	PROPN
ejpam-4412	658	20	,	,	PUNCT
ejpam-4412	658	21	for	for	ADP
ejpam-4412	658	22	some	some	DET
ejpam-4412	658	23	log	log	NOUN
ejpam-4412	658	24	-	-	PUNCT
ejpam-4412	658	25	hyponormal	hyponormal	NOUN
ejpam-4412	658	26	t	t	PROPN
ejpam-4412	658	27	∗	∗	X
ejpam-4412	658	28	4	4	NUM
ejpam-4412	658	29	.	.	PUNCT
ejpam-4412	659	1	let	let	VERB
ejpam-4412	659	2	x	x	PUNCT
ejpam-4412	659	3	=	=	PRON
ejpam-4412	659	4	(	(	PUNCT
ejpam-4412	659	5	x11	x11	NOUN
ejpam-4412	659	6	x12	x12	NUM
ejpam-4412	659	7	x21	x21	NUM
ejpam-4412	659	8	x22	x22	PROPN
ejpam-4412	659	9	)	)	PUNCT
ejpam-4412	659	10	.	.	PUNCT
ejpam-4412	660	1	then	then	ADV
ejpam-4412	660	2	x	x	X
ejpam-4412	660	3	′	′	NOUN
ejpam-4412	660	4	=	=	SYM
ejpam-4412	660	5	xu	xu	PROPN
ejpam-4412	660	6	|t	|t	PROPN
ejpam-4412	661	1	|t	|t	PROPN
ejpam-4412	661	2	implies	imply	VERB
ejpam-4412	661	3	that	that	SCONJ
ejpam-4412	661	4	x12	x12	NUM
ejpam-4412	661	5	=	=	SYM
ejpam-4412	661	6	0	0	NUM
ejpam-4412	661	7	,	,	PUNCT
ejpam-4412	661	8	x21	x21	PROPN
ejpam-4412	661	9	=	=	SYM
ejpam-4412	661	10	0	0	NUM
ejpam-4412	661	11	and	and	CCONJ
ejpam-4412	661	12	x22	x22	NUM
ejpam-4412	662	1	=	=	SYM
ejpam-4412	662	2	0	0	PROPN
ejpam-4412	662	3	.	.	PUNCT
ejpam-4412	663	1	the	the	DET
ejpam-4412	663	2	assumption	assumption	NOUN
ejpam-4412	663	3	xt	xt	PUNCT
ejpam-4412	664	1	=	=	SYM
ejpam-4412	664	2	sx	sx	PROPN
ejpam-4412	664	3	implies	imply	VERB
ejpam-4412	664	4	that	that	SCONJ
ejpam-4412	664	5	x11t1	x11t1	PROPN
ejpam-4412	664	6	=	=	SYM
ejpam-4412	664	7	s1x11	s1x11	NOUN
ejpam-4412	664	8	,	,	PUNCT
ejpam-4412	664	9	hence	hence	ADV
ejpam-4412	664	10	x11	x11	PROPN
ejpam-4412	664	11	t	t	PROPN
ejpam-4412	664	12	∗	∗	X
ejpam-4412	664	13	1	1	NUM
ejpam-4412	664	14	⊕	⊕	NUM
ejpam-4412	664	15	0	0	NUM
ejpam-4412	665	1	=	=	AUX
ejpam-4412	665	2	s∗	s∗	PROPN
ejpam-4412	665	3	1x11	1x11	NUM
ejpam-4412	665	4	by	by	ADP
ejpam-4412	665	5	fuglede	fuglede	PROPN
ejpam-4412	665	6	-	-	PUNCT
ejpam-4412	665	7	putnam	putnam	NOUN
ejpam-4412	665	8	theorem	theorem	NOUN
ejpam-4412	665	9	.	.	PUNCT
ejpam-4412	666	1	thus	thus	ADV
ejpam-4412	666	2	we	we	PRON
ejpam-4412	666	3	have	have	VERB
ejpam-4412	666	4	xt	xt	X
ejpam-4412	666	5	∗	∗	NOUN
ejpam-4412	666	6	=	=	SYM
ejpam-4412	666	7	x11	x11	PROPN
ejpam-4412	666	8	t	t	PROPN
ejpam-4412	666	9	∗	∗	X
ejpam-4412	666	10	1	1	NUM
ejpam-4412	666	11	⊕	⊕	NUM
ejpam-4412	666	12	0	0	NUM
ejpam-4412	667	1	=	=	PUNCT
ejpam-4412	667	2	s∗	s∗	PROPN
ejpam-4412	667	3	1x11	1x11	NUM
ejpam-4412	667	4	⊕	⊕	PROPN
ejpam-4412	667	5	0	0	NUM
ejpam-4412	668	1	=	=	SYM
ejpam-4412	668	2	s∗x	s∗x	PROPN
ejpam-4412	668	3	.	.	PUNCT
ejpam-4412	669	1	therefore	therefore	ADV
ejpam-4412	669	2	,	,	PUNCT
ejpam-4412	669	3	the	the	DET
ejpam-4412	669	4	proof	proof	NOUN
ejpam-4412	669	5	of	of	ADP
ejpam-4412	669	6	the	the	DET
ejpam-4412	669	7	theorem	theorem	NOUN
ejpam-4412	669	8	is	be	AUX
ejpam-4412	669	9	achieved	achieve	VERB
ejpam-4412	669	10	.	.	PUNCT
ejpam-4412	670	1	example	example	NOUN
ejpam-4412	671	1	4	4	NUM
ejpam-4412	671	2	.	.	PUNCT
ejpam-4412	672	1	let	let	VERB
ejpam-4412	672	2	r	r	PRON
ejpam-4412	672	3	be	be	AUX
ejpam-4412	672	4	an	an	DET
ejpam-4412	672	5	operator	operator	NOUN
ejpam-4412	672	6	such	such	ADJ
ejpam-4412	672	7	that	that	DET
ejpam-4412	672	8	ker(r	ker(r	PROPN
ejpam-4412	672	9	)	)	PUNCT
ejpam-4412	672	10	does	do	AUX
ejpam-4412	672	11	not	not	PART
ejpam-4412	672	12	reduce	reduce	VERB
ejpam-4412	672	13	r	r	NOUN
ejpam-4412	672	14	and	and	CCONJ
ejpam-4412	672	15	let	let	VERB
ejpam-4412	672	16	p	p	PRON
ejpam-4412	672	17	be	be	AUX
ejpam-4412	672	18	the	the	DET
ejpam-4412	672	19	orthogonal	orthogonal	ADJ
ejpam-4412	672	20	projection	projection	NOUN
ejpam-4412	672	21	onto	onto	ADP
ejpam-4412	672	22	ker(r	ker(r	PROPN
ejpam-4412	672	23	)	)	PUNCT
ejpam-4412	672	24	.	.	PUNCT
ejpam-4412	673	1	then	then	ADV
ejpam-4412	673	2	p	p	NOUN
ejpam-4412	673	3	does	do	AUX
ejpam-4412	673	4	not	not	PART
ejpam-4412	673	5	commute	commute	VERB
ejpam-4412	673	6	with	with	ADP
ejpam-4412	673	7	t	t	NOUN
ejpam-4412	673	8	;	;	PUNCT
ejpam-4412	673	9	otherwise	otherwise	ADV
ejpam-4412	673	10	ran(r	ran(r	NUM
ejpam-4412	673	11	)	)	PUNCT
ejpam-4412	673	12	=	=	SYM
ejpam-4412	673	13	ker(r	ker(r	PROPN
ejpam-4412	673	14	)	)	PUNCT
ejpam-4412	673	15	reduce	reduce	VERB
ejpam-4412	673	16	t	t	PROPN
ejpam-4412	673	17	.	.	PUNCT
ejpam-4412	674	1	hence	hence	ADV
ejpam-4412	674	2	pr	pr	VERB
ejpam-4412	674	3	̸=	̸=	PROPN
ejpam-4412	674	4	0	0	NUM
ejpam-4412	674	5	=	=	PUNCT
ejpam-4412	674	6	rp	rp	NOUN
ejpam-4412	674	7	.	.	PUNCT
ejpam-4412	675	1	it	it	PRON
ejpam-4412	675	2	is	be	AUX
ejpam-4412	675	3	easy	easy	ADJ
ejpam-4412	675	4	to	to	PART
ejpam-4412	675	5	see	see	VERB
ejpam-4412	675	6	that	that	DET
ejpam-4412	675	7	rp	rp	NOUN
ejpam-4412	675	8	=	=	PUNCT
ejpam-4412	675	9	pr∗	pr∗	NOUN
ejpam-4412	675	10	=	=	SYM
ejpam-4412	675	11	0	0	NUM
ejpam-4412	676	1	but	but	CCONJ
ejpam-4412	676	2	r∗p	r∗p	NUM
ejpam-4412	676	3	̸=	̸=	PROPN
ejpam-4412	676	4	pr(̸=	pr(̸=	NOUN
ejpam-4412	676	5	0	0	NUM
ejpam-4412	676	6	)	)	PUNCT
ejpam-4412	676	7	because	because	SCONJ
ejpam-4412	676	8	ran(r∗p	ran(r∗p	PROPN
ejpam-4412	676	9	)	)	PUNCT
ejpam-4412	676	10	⊂	⊂	NOUN
ejpam-4412	676	11	ran(r∗	ran(r∗	NOUN
ejpam-4412	676	12	)	)	PUNCT
ejpam-4412	676	13	⊂	⊂	PROPN
ejpam-4412	676	14	ker(r⊥	ker(r⊥	PROPN
ejpam-4412	676	15	)	)	PUNCT
ejpam-4412	677	1	=	=	PUNCT
ejpam-4412	678	1	i	i	PRON
ejpam-4412	678	2	−	−	PROPN
ejpam-4412	679	1	p.	p.	NOUN
ejpam-4412	679	2	if	if	SCONJ
ejpam-4412	679	3	we	we	PRON
ejpam-4412	679	4	put	put	VERB
ejpam-4412	679	5	t	t	NOUN
ejpam-4412	679	6	=	=	SYM
ejpam-4412	679	7	r	r	NOUN
ejpam-4412	679	8	,	,	PUNCT
ejpam-4412	679	9	then	then	ADV
ejpam-4412	679	10	the	the	DET
ejpam-4412	679	11	assertion	assertion	NOUN
ejpam-4412	679	12	of	of	ADP
ejpam-4412	679	13	theorem	theorem	NOUN
ejpam-4412	679	14	10	10	NUM
ejpam-4412	679	15	does	do	AUX
ejpam-4412	679	16	not	not	PART
ejpam-4412	679	17	hold	hold	VERB
ejpam-4412	679	18	for	for	ADP
ejpam-4412	679	19	such	such	ADJ
ejpam-4412	679	20	t	t	NOUN
ejpam-4412	679	21	.	.	PUNCT
ejpam-4412	680	1	also	also	ADV
ejpam-4412	680	2	,	,	PUNCT
ejpam-4412	680	3	if	if	SCONJ
ejpam-4412	680	4	we	we	PRON
ejpam-4412	680	5	put	put	VERB
ejpam-4412	680	6	t	t	NOUN
ejpam-4412	680	7	=	=	SYM
ejpam-4412	680	8	r∗	r∗	PROPN
ejpam-4412	680	9	,	,	PUNCT
ejpam-4412	680	10	s	s	PART
ejpam-4412	680	11	=	=	PUNCT
ejpam-4412	680	12	i	i	PRON
ejpam-4412	680	13	−p	−p	ADJ
ejpam-4412	680	14	and	and	CCONJ
ejpam-4412	680	15	x	x	X
ejpam-4412	680	16	=	=	SYM
ejpam-4412	680	17	p	p	X
ejpam-4412	680	18	,	,	PUNCT
ejpam-4412	680	19	then	then	ADV
ejpam-4412	680	20	xt	xt	X
ejpam-4412	680	21	=	=	SYM
ejpam-4412	680	22	pr∗	pr∗	PROPN
ejpam-4412	680	23	=	=	SYM
ejpam-4412	680	24	0	0	PUNCT
ejpam-4412	681	1	=	=	SYM
ejpam-4412	681	2	(	(	PUNCT
ejpam-4412	681	3	i	i	PRON
ejpam-4412	681	4	−	−	PROPN
ejpam-4412	682	1	p	p	NOUN
ejpam-4412	682	2	)	)	PUNCT
ejpam-4412	682	3	p	p	X
ejpam-4412	682	4	=	=	PUNCT
ejpam-4412	682	5	sx	sx	PROPN
ejpam-4412	682	6	.	.	PUNCT
ejpam-4412	683	1	however	however	ADV
ejpam-4412	683	2	,	,	PUNCT
ejpam-4412	683	3	xt	xt	X
ejpam-4412	683	4	∗	∗	NOUN
ejpam-4412	683	5	=	=	PUNCT
ejpam-4412	683	6	pr	pr	X
ejpam-4412	683	7	̸=	̸=	PROPN
ejpam-4412	683	8	0	0	NUM
ejpam-4412	684	1	=	=	SYM
ejpam-4412	684	2	(	(	PUNCT
ejpam-4412	684	3	i	i	PRON
ejpam-4412	684	4	−	−	PROPN
ejpam-4412	685	1	p	p	NOUN
ejpam-4412	685	2	)	)	PUNCT
ejpam-4412	685	3	p	p	NOUN
ejpam-4412	685	4	=	=	NOUN
ejpam-4412	685	5	s∗x	s∗x	PROPN
ejpam-4412	685	6	.	.	NOUN
ejpam-4412	685	7	hence	hence	ADV
ejpam-4412	685	8	the	the	DET
ejpam-4412	685	9	assertion	assertion	NOUN
ejpam-4412	685	10	of	of	ADP
ejpam-4412	685	11	theorem	theorem	NOUN
ejpam-4412	685	12	12	12	NUM
ejpam-4412	685	13	does	do	AUX
ejpam-4412	685	14	not	not	PART
ejpam-4412	685	15	hold	hold	VERB
ejpam-4412	685	16	for	for	ADP
ejpam-4412	685	17	such	such	ADJ
ejpam-4412	685	18	t	t	NOUN
ejpam-4412	685	19	.	.	PUNCT
ejpam-4412	686	1	references	reference	NOUN
ejpam-4412	686	2	1087	1087	NUM
ejpam-4412	686	3	theorem	theorem	VERB
ejpam-4412	686	4	13	13	NUM
ejpam-4412	686	5	.	.	PUNCT
ejpam-4412	687	1	let	let	AUX
ejpam-4412	687	2	t	t	PROPN
ejpam-4412	687	3	∈	∈	PROPN
ejpam-4412	687	4	b(h	b(h	PROPN
ejpam-4412	687	5	)	)	PUNCT
ejpam-4412	687	6	be	be	VERB
ejpam-4412	687	7	such	such	ADJ
ejpam-4412	687	8	that	that	SCONJ
ejpam-4412	687	9	t	t	PROPN
ejpam-4412	687	10	∗	∗	NOUN
ejpam-4412	687	11	is	be	AUX
ejpam-4412	687	12	an	an	DET
ejpam-4412	687	13	injective	injective	ADJ
ejpam-4412	687	14	class	class	NOUN
ejpam-4412	687	15	p	p	NOUN
ejpam-4412	687	16	-	-	PUNCT
ejpam-4412	687	17	wa(s	wa(s	NUM
ejpam-4412	687	18	,	,	PUNCT
ejpam-4412	687	19	t	t	PROPN
ejpam-4412	687	20	)	)	PUNCT
ejpam-4412	687	21	for	for	ADP
ejpam-4412	687	22	0	0	NUM
ejpam-4412	687	23	<	<	X
ejpam-4412	687	24	s	s	PROPN
ejpam-4412	687	25	,	,	PUNCT
ejpam-4412	687	26	t	t	PROPN
ejpam-4412	687	27	,	,	PUNCT
ejpam-4412	687	28	s	s	PART
ejpam-4412	687	29	+	+	X
ejpam-4412	687	30	t	t	NOUN
ejpam-4412	687	31	=	=	PUNCT
ejpam-4412	687	32	and	and	CCONJ
ejpam-4412	687	33	0	0	NUM
ejpam-4412	687	34	<	<	X
ejpam-4412	687	35	p	p	X
ejpam-4412	687	36	≤	≤	NUM
ejpam-4412	687	37	1	1	NUM
ejpam-4412	687	38	.	.	PUNCT
ejpam-4412	688	1	let	let	VERB
ejpam-4412	688	2	s	s	PRON
ejpam-4412	688	3	∈	∈	PROPN
ejpam-4412	688	4	b(k	b(k	PROPN
ejpam-4412	688	5	)	)	PUNCT
ejpam-4412	689	1	be	be	AUX
ejpam-4412	689	2	dominant	dominant	ADJ
ejpam-4412	689	3	.	.	PUNCT
ejpam-4412	690	1	if	if	SCONJ
ejpam-4412	690	2	xt	xt	PROPN
ejpam-4412	690	3	=	=	SYM
ejpam-4412	690	4	sx	sx	PROPN
ejpam-4412	690	5	,	,	PUNCT
ejpam-4412	690	6	for	for	ADP
ejpam-4412	690	7	some	some	DET
ejpam-4412	690	8	x	x	SYM
ejpam-4412	690	9	∈	∈	PROPN
ejpam-4412	690	10	b(h	b(h	PROPN
ejpam-4412	690	11	,	,	PUNCT
ejpam-4412	690	12	k	k	NOUN
ejpam-4412	690	13	)	)	PUNCT
ejpam-4412	690	14	.	.	PUNCT
ejpam-4412	691	1	then	then	ADV
ejpam-4412	691	2	xt	xt	ADP
ejpam-4412	691	3	∗	∗	NOUN
ejpam-4412	691	4	=	=	SYM
ejpam-4412	691	5	s∗x	s∗x	NOUN
ejpam-4412	691	6	.	.	PUNCT
ejpam-4412	691	7	proof	proof	NOUN
ejpam-4412	691	8	.	.	PUNCT
ejpam-4412	692	1	assume	assume	VERB
ejpam-4412	692	2	that	that	SCONJ
ejpam-4412	692	3	t	t	PROPN
ejpam-4412	692	4	∗	∗	NOUN
ejpam-4412	692	5	is	be	AUX
ejpam-4412	692	6	an	an	DET
ejpam-4412	692	7	injective	injective	ADJ
ejpam-4412	692	8	p	p	NOUN
ejpam-4412	692	9	-	-	PUNCT
ejpam-4412	692	10	w	w	NOUN
ejpam-4412	692	11	-	-	PUNCT
ejpam-4412	692	12	hyponormal	hyponormal	ADJ
ejpam-4412	692	13	and	and	CCONJ
ejpam-4412	692	14	let	let	VERB
ejpam-4412	692	15	t	t	NOUN
ejpam-4412	692	16	=	=	SYM
ejpam-4412	692	17	u	u	NOUN
ejpam-4412	692	18	|t	|t	VERB
ejpam-4412	692	19	|	|	ADV
ejpam-4412	692	20	be	be	AUX
ejpam-4412	692	21	the	the	DET
ejpam-4412	692	22	polar	polar	ADJ
ejpam-4412	692	23	decomposition	decomposition	NOUN
ejpam-4412	692	24	of	of	ADP
ejpam-4412	692	25	t	t	PROPN
ejpam-4412	692	26	.	.	PUNCT
ejpam-4412	693	1	let	let	VERB
ejpam-4412	693	2	t	t	PROPN
ejpam-4412	693	3	(	(	PUNCT
ejpam-4412	693	4	s	s	PROPN
ejpam-4412	693	5	,	,	PUNCT
ejpam-4412	693	6	t	t	PROPN
ejpam-4412	693	7	)	)	PUNCT
ejpam-4412	693	8	be	be	VERB
ejpam-4412	693	9	the	the	DET
ejpam-4412	693	10	aluthge	aluthge	ADJ
ejpam-4412	693	11	transform	transform	NOUN
ejpam-4412	693	12	of	of	ADP
ejpam-4412	693	13	t	t	PROPN
ejpam-4412	693	14	and	and	CCONJ
ejpam-4412	693	15	x	x	SYM
ejpam-4412	693	16	′	′	NUM
ejpam-4412	693	17	=	=	SYM
ejpam-4412	693	18	xu	xu	PROPN
ejpam-4412	693	19	|t	|t	PROPN
ejpam-4412	694	1	|t	|t	PROPN
ejpam-4412	694	2	.	.	PUNCT
ejpam-4412	695	1	then	then	ADV
ejpam-4412	695	2	x	x	SYM
ejpam-4412	695	3	′t	′t	NOUN
ejpam-4412	695	4	(	(	PUNCT
ejpam-4412	695	5	s	s	PROPN
ejpam-4412	695	6	,	,	PUNCT
ejpam-4412	695	7	t	t	PROPN
ejpam-4412	695	8	)	)	PUNCT
ejpam-4412	695	9	=	=	VERB
ejpam-4412	695	10	sx	sx	INTJ
ejpam-4412	695	11	′	′	NOUN
ejpam-4412	696	1	and	and	CCONJ
ejpam-4412	696	2	t	t	PROPN
ejpam-4412	696	3	∗(s	∗(s	PROPN
ejpam-4412	696	4	,	,	PUNCT
ejpam-4412	696	5	t	t	PROPN
ejpam-4412	696	6	)	)	PUNCT
ejpam-4412	696	7	is	be	AUX
ejpam-4412	696	8	rp	rp	NOUN
ejpam-4412	696	9	-	-	PUNCT
ejpam-4412	696	10	hyponormal	hyponormal	ADJ
ejpam-4412	696	11	and	and	CCONJ
ejpam-4412	696	12	satisfies	satisfie	NOUN
ejpam-4412	696	13	|t	|t	VERB
ejpam-4412	696	14	∗(s	∗(s	PROPN
ejpam-4412	696	15	,	,	PUNCT
ejpam-4412	696	16	t)|2rp	t)|2rp	VERB
ejpam-4412	696	17	≥	≥	NOUN
ejpam-4412	696	18	|t	|t	VERB
ejpam-4412	696	19	∗|2rp	∗|2rp	ADJ
ejpam-4412	696	20	≥	≥	NUM
ejpam-4412	696	21	|(t	|(t	NOUN
ejpam-4412	696	22	∗(s	∗(s	NOUN
ejpam-4412	696	23	,	,	PUNCT
ejpam-4412	696	24	t))∗|2rp	t))∗|2rp	VERB
ejpam-4412	696	25	for	for	ADP
ejpam-4412	696	26	r	r	NOUN
ejpam-4412	696	27	∈	∈	PROPN
ejpam-4412	696	28	min{s	min{s	PROPN
ejpam-4412	696	29	,	,	PUNCT
ejpam-4412	696	30	t	t	PROPN
ejpam-4412	696	31	}	}	PUNCT
ejpam-4412	696	32	.	.	PUNCT
ejpam-4412	697	1	by	by	ADP
ejpam-4412	697	2	the	the	DET
ejpam-4412	697	3	same	same	ADJ
ejpam-4412	697	4	argument	argument	NOUN
ejpam-4412	697	5	in	in	ADP
ejpam-4412	697	6	the	the	DET
ejpam-4412	697	7	proof	proof	NOUN
ejpam-4412	697	8	of	of	ADP
ejpam-4412	697	9	theorem	theorem	NOUN
ejpam-4412	697	10	12	12	NUM
ejpam-4412	697	11	,	,	PUNCT
ejpam-4412	697	12	we	we	PRON
ejpam-4412	697	13	conclude	conclude	VERB
ejpam-4412	697	14	that	that	DET
ejpam-4412	697	15	t	t	PROPN
ejpam-4412	697	16	∗(s	∗(s	PROPN
ejpam-4412	697	17	,	,	PUNCT
ejpam-4412	697	18	t	t	PROPN
ejpam-4412	697	19	)	)	PUNCT
ejpam-4412	697	20	=	=	PROPN
ejpam-4412	697	21	t1	t1	PROPN
ejpam-4412	697	22	⊕	⊕	PROPN
ejpam-4412	697	23	t3	t3	PROPN
ejpam-4412	697	24	on	on	ADP
ejpam-4412	697	25	h	h	NOUN
ejpam-4412	697	26	=	=	SYM
ejpam-4412	697	27	ker(x	ker(x	PROPN
ejpam-4412	697	28	′)⊥	′)⊥	PROPN
ejpam-4412	697	29	⊕	⊕	PROPN
ejpam-4412	697	30	ker(x	ker(x	PROPN
ejpam-4412	697	31	′	′	NOUN
ejpam-4412	697	32	)	)	PUNCT
ejpam-4412	697	33	and	and	CCONJ
ejpam-4412	697	34	s	s	NOUN
ejpam-4412	697	35	=	=	SYM
ejpam-4412	697	36	s1	s1	PROPN
ejpam-4412	697	37	⊕	⊕	PROPN
ejpam-4412	697	38	s3	s3	PROPN
ejpam-4412	697	39	,	,	PUNCT
ejpam-4412	697	40	where	where	SCONJ
ejpam-4412	697	41	t1	t1	NOUN
ejpam-4412	697	42	is	be	AUX
ejpam-4412	697	43	an	an	DET
ejpam-4412	697	44	injective	injective	ADJ
ejpam-4412	697	45	normal	normal	ADJ
ejpam-4412	697	46	operator	operator	NOUN
ejpam-4412	697	47	and	and	CCONJ
ejpam-4412	697	48	s1	s1	NOUN
ejpam-4412	697	49	is	be	AUX
ejpam-4412	697	50	also	also	ADV
ejpam-4412	697	51	normal	normal	ADJ
ejpam-4412	697	52	,	,	PUNCT
ejpam-4412	697	53	t	t	PROPN
ejpam-4412	697	54	∗	∗	NOUN
ejpam-4412	697	55	3	3	NUM
ejpam-4412	697	56	is	be	AUX
ejpam-4412	697	57	invertible	invertible	ADJ
ejpam-4412	697	58	class	class	NOUN
ejpam-4412	697	59	p	p	NOUN
ejpam-4412	697	60	-	-	PUNCT
ejpam-4412	697	61	wa(s	wa(s	NUM
ejpam-4412	697	62	,	,	PUNCT
ejpam-4412	697	63	t	t	PROPN
ejpam-4412	697	64	)	)	PUNCT
ejpam-4412	697	65	and	and	CCONJ
ejpam-4412	697	66	s3	s3	PROPN
ejpam-4412	697	67	is	be	AUX
ejpam-4412	697	68	dominant	dominant	ADJ
ejpam-4412	697	69	.	.	PUNCT
ejpam-4412	698	1	hence	hence	ADV
ejpam-4412	698	2	by	by	ADP
ejpam-4412	698	3	lemma	lemma	PROPN
ejpam-4412	698	4	7	7	NUM
ejpam-4412	698	5	,	,	PUNCT
ejpam-4412	698	6	we	we	PRON
ejpam-4412	698	7	have	have	VERB
ejpam-4412	698	8	that	that	DET
ejpam-4412	698	9	t	t	PROPN
ejpam-4412	698	10	is	be	AUX
ejpam-4412	698	11	of	of	ADP
ejpam-4412	698	12	the	the	DET
ejpam-4412	698	13	form	form	NOUN
ejpam-4412	698	14	t	t	PROPN
ejpam-4412	698	15	=	=	SYM
ejpam-4412	698	16	t1	t1	PROPN
ejpam-4412	698	17	⊕	⊕	PROPN
ejpam-4412	698	18	t4	t4	PROPN
ejpam-4412	698	19	for	for	ADP
ejpam-4412	698	20	some	some	DET
ejpam-4412	698	21	class	class	NOUN
ejpam-4412	698	22	p	p	NOUN
ejpam-4412	698	23	-	-	PUNCT
ejpam-4412	698	24	wa(s	wa(s	NUM
ejpam-4412	698	25	,	,	PUNCT
ejpam-4412	698	26	t	t	PROPN
ejpam-4412	698	27	)	)	PUNCT
ejpam-4412	698	28	t	t	PROPN
ejpam-4412	698	29	∗	∗	NOUN
ejpam-4412	698	30	4	4	NUM
ejpam-4412	698	31	.	.	PUNCT
ejpam-4412	699	1	let	let	VERB
ejpam-4412	699	2	x	x	PUNCT
ejpam-4412	699	3	=	=	PRON
ejpam-4412	699	4	(	(	PUNCT
ejpam-4412	699	5	x11	x11	NOUN
ejpam-4412	699	6	x12	x12	NUM
ejpam-4412	699	7	x21	x21	NUM
ejpam-4412	699	8	x22	x22	PROPN
ejpam-4412	699	9	)	)	PUNCT
ejpam-4412	699	10	.	.	PUNCT
ejpam-4412	700	1	then	then	ADV
ejpam-4412	700	2	x	x	X
ejpam-4412	700	3	′	′	NOUN
ejpam-4412	700	4	=	=	SYM
ejpam-4412	700	5	xu	xu	PROPN
ejpam-4412	700	6	|t	|t	PROPN
ejpam-4412	701	1	|t	|t	PROPN
ejpam-4412	701	2	implies	imply	VERB
ejpam-4412	701	3	that	that	SCONJ
ejpam-4412	701	4	x12	x12	NUM
ejpam-4412	701	5	=	=	SYM
ejpam-4412	701	6	0	0	NUM
ejpam-4412	701	7	,	,	PUNCT
ejpam-4412	701	8	x21	x21	PROPN
ejpam-4412	701	9	=	=	SYM
ejpam-4412	701	10	0	0	NUM
ejpam-4412	701	11	and	and	CCONJ
ejpam-4412	701	12	x22	x22	NUM
ejpam-4412	702	1	=	=	SYM
ejpam-4412	702	2	0	0	PROPN
ejpam-4412	702	3	.	.	PUNCT
ejpam-4412	703	1	the	the	DET
ejpam-4412	703	2	assumption	assumption	NOUN
ejpam-4412	703	3	xt	xt	PUNCT
ejpam-4412	704	1	=	=	SYM
ejpam-4412	704	2	sx	sx	PROPN
ejpam-4412	704	3	implies	imply	VERB
ejpam-4412	704	4	that	that	SCONJ
ejpam-4412	704	5	x11t1	x11t1	PROPN
ejpam-4412	704	6	=	=	SYM
ejpam-4412	704	7	s1x11	s1x11	NOUN
ejpam-4412	704	8	,	,	PUNCT
ejpam-4412	704	9	hence	hence	ADV
ejpam-4412	704	10	x11	x11	PROPN
ejpam-4412	704	11	t	t	PROPN
ejpam-4412	704	12	∗	∗	X
ejpam-4412	704	13	1	1	NUM
ejpam-4412	704	14	=	=	SYM
ejpam-4412	704	15	s∗	s∗	PROPN
ejpam-4412	704	16	1x11	1x11	NUM
ejpam-4412	704	17	by	by	ADP
ejpam-4412	704	18	fuglede	fuglede	PROPN
ejpam-4412	704	19	-	-	PUNCT
ejpam-4412	704	20	putnam	putnam	NOUN
ejpam-4412	704	21	theorem	theorem	NOUN
ejpam-4412	704	22	.	.	PUNCT
ejpam-4412	705	1	thus	thus	ADV
ejpam-4412	705	2	we	we	PRON
ejpam-4412	705	3	have	have	VERB
ejpam-4412	705	4	xt	xt	X
ejpam-4412	705	5	∗	∗	NOUN
ejpam-4412	705	6	=	=	SYM
ejpam-4412	705	7	x11	x11	PROPN
ejpam-4412	705	8	t	t	PROPN
ejpam-4412	705	9	∗	∗	X
ejpam-4412	705	10	1	1	NUM
ejpam-4412	705	11	⊕	⊕	NUM
ejpam-4412	705	12	0	0	NUM
ejpam-4412	706	1	=	=	PUNCT
ejpam-4412	706	2	s∗	s∗	PROPN
ejpam-4412	706	3	1x11	1x11	NUM
ejpam-4412	706	4	⊕	⊕	PROPN
ejpam-4412	706	5	0	0	NUM
ejpam-4412	707	1	=	=	SYM
ejpam-4412	707	2	s∗x	s∗x	PROPN
ejpam-4412	707	3	.	.	PUNCT
ejpam-4412	708	1	therefore	therefore	ADV
ejpam-4412	708	2	,	,	PUNCT
ejpam-4412	708	3	the	the	DET
ejpam-4412	708	4	proof	proof	NOUN
ejpam-4412	708	5	of	of	ADP
ejpam-4412	708	6	the	the	DET
ejpam-4412	708	7	theorem	theorem	NOUN
ejpam-4412	708	8	is	be	AUX
ejpam-4412	708	9	achieved	achieve	VERB
ejpam-4412	708	10	.	.	PUNCT
ejpam-4412	709	1	example	example	NOUN
ejpam-4412	709	2	5	5	NUM
ejpam-4412	709	3	.	.	PUNCT
ejpam-4412	710	1	let	let	VERB
ejpam-4412	710	2	t	t	PROPN
ejpam-4412	710	3	∗	∗	NOUN
ejpam-4412	710	4	=	=	PUNCT
ejpam-4412	710	5	r	r	NOUN
ejpam-4412	710	6	as	as	ADP
ejpam-4412	710	7	in	in	ADP
ejpam-4412	710	8	example	example	NOUN
ejpam-4412	710	9	2	2	X
ejpam-4412	710	10	.	.	PUNCT
ejpam-4412	711	1	let	let	VERB
ejpam-4412	711	2	x	x	PUNCT
ejpam-4412	711	3	=	=	PUNCT
ejpam-4412	711	4	p	p	NOUN
ejpam-4412	711	5	be	be	AUX
ejpam-4412	711	6	the	the	DET
ejpam-4412	711	7	orthogonal	orthogonal	ADJ
ejpam-4412	711	8	projection	projection	NOUN
ejpam-4412	711	9	onto	onto	ADP
ejpam-4412	711	10	ker(t	ker(t	NOUN
ejpam-4412	711	11	∗	∗	NOUN
ejpam-4412	711	12	)	)	PUNCT
ejpam-4412	711	13	and	and	CCONJ
ejpam-4412	711	14	s	s	VERB
ejpam-4412	711	15	=	=	NOUN
ejpam-4412	712	1	i	i	PRON
ejpam-4412	712	2	−	−	PROPN
ejpam-4412	713	1	p.	p.	NOUN
ejpam-4412	713	2	then	then	ADV
ejpam-4412	713	3	sx	sx	PROPN
ejpam-4412	713	4	=	=	SYM
ejpam-4412	713	5	0	0	PUNCT
ejpam-4412	713	6	=	=	SYM
ejpam-4412	713	7	xt	xt	ADP
ejpam-4412	713	8	∗	∗	NOUN
ejpam-4412	713	9	,	,	PUNCT
ejpam-4412	713	10	but	but	CCONJ
ejpam-4412	713	11	0	0	X
ejpam-4412	713	12	=	=	SYM
ejpam-4412	713	13	s∗x	s∗x	VERB
ejpam-4412	713	14	̸=	̸=	PROPN
ejpam-4412	713	15	xt	xt	ADP
ejpam-4412	714	1	∗.	∗.	PROPN
ejpam-4412	714	2	hence	hence	ADV
ejpam-4412	714	3	the	the	DET
ejpam-4412	714	4	injectivity	injectivity	NOUN
ejpam-4412	714	5	condition	condition	NOUN
ejpam-4412	714	6	is	be	AUX
ejpam-4412	714	7	necessary	necessary	ADJ
ejpam-4412	714	8	for	for	ADP
ejpam-4412	714	9	theorem	theorem	ADJ
ejpam-4412	714	10	13	13	NUM
ejpam-4412	714	11	.	.	PUNCT
ejpam-4412	715	1	references	reference	NOUN
ejpam-4412	715	2	[	[	X
ejpam-4412	715	3	1	1	NUM
ejpam-4412	715	4	]	]	PUNCT
ejpam-4412	715	5	a.	a.	NOUN
ejpam-4412	715	6	uchiyama	uchiyama	NOUN
ejpam-4412	715	7	,	,	PUNCT
ejpam-4412	715	8	k.	k.	PROPN
ejpam-4412	715	9	tanahashi	tanahashi	PROPN
ejpam-4412	715	10	and	and	CCONJ
ejpam-4412	715	11	j.	j.	PROPN
ejpam-4412	715	12	i.	i.	PROPN
ejpam-4412	715	13	lee	lee	PROPN
ejpam-4412	715	14	.	.	PROPN
ejpam-4412	716	1	spectrum	spectrum	PROPN
ejpam-4412	716	2	of	of	ADP
ejpam-4412	716	3	class	class	NOUN
ejpam-4412	716	4	a(s	a(s	PROPN
ejpam-4412	716	5	,	,	PUNCT
ejpam-4412	716	6	t	t	PROPN
ejpam-4412	716	7	)	)	PUNCT
ejpam-4412	716	8	operators	operator	NOUN
ejpam-4412	716	9	.	.	PUNCT
ejpam-4412	717	1	acta	acta	PROPN
ejpam-4412	717	2	sci	sci	PROPN
ejpam-4412	717	3	.	.	PROPN
ejpam-4412	717	4	math	math	PROPN
ejpam-4412	717	5	.	.	PUNCT
ejpam-4412	718	1	(	(	PUNCT
ejpam-4412	718	2	szeged	szeged	PROPN
ejpam-4412	718	3	)	)	PUNCT
ejpam-4412	718	4	,	,	PUNCT
ejpam-4412	718	5	70:279–287	70:279–287	NUM
ejpam-4412	718	6	,	,	PUNCT
ejpam-4412	718	7	2004	2004	NUM
ejpam-4412	718	8	.	.	PUNCT
ejpam-4412	719	1	[	[	X
ejpam-4412	719	2	2	2	NUM
ejpam-4412	719	3	]	]	PUNCT
ejpam-4412	719	4	a.	a.	NOUN
ejpam-4412	719	5	aluthge	aluthge	PROPN
ejpam-4412	719	6	.	.	PUNCT
ejpam-4412	720	1	on	on	ADP
ejpam-4412	720	2	p	p	PROPN
ejpam-4412	720	3	-	-	PUNCT
ejpam-4412	720	4	hyponormal	hyponormal	ADJ
ejpam-4412	720	5	operators	operator	NOUN
ejpam-4412	720	6	for	for	ADP
ejpam-4412	720	7	0	0	NUM
ejpam-4412	720	8	<	<	X
ejpam-4412	720	9	p	p	X
ejpam-4412	720	10	<	<	X
ejpam-4412	720	11	1	1	NUM
ejpam-4412	720	12	.	.	PUNCT
ejpam-4412	720	13	integral	integral	ADJ
ejpam-4412	720	14	equations	equation	NOUN
ejpam-4412	720	15	operator	operator	NOUN
ejpam-4412	720	16	theory	theory	NOUN
ejpam-4412	720	17	,	,	PUNCT
ejpam-4412	720	18	13:307–315	13:307–315	NUM
ejpam-4412	720	19	,	,	PUNCT
ejpam-4412	720	20	1990	1990	NUM
ejpam-4412	720	21	.	.	PUNCT
ejpam-4412	721	1	[	[	X
ejpam-4412	721	2	3	3	X
ejpam-4412	721	3	]	]	X
ejpam-4412	721	4	ž.	ž.	NOUN
ejpam-4412	721	5	mijajlovič	mijajlovič	INTJ
ejpam-4412	721	6	et	et	PROPN
ejpam-4412	721	7	al	al	PROPN
ejpam-4412	721	8	.	.	PROPN
ejpam-4412	721	9	nestandardna	nestandardna	PROPN
ejpam-4412	721	10	analiza	analiza	PROPN
ejpam-4412	721	11	.	.	PUNCT
ejpam-4412	722	1	univesity	univesity	PROPN
ejpam-4412	722	2	of	of	ADP
ejpam-4412	722	3	belgrade	belgrade	PROPN
ejpam-4412	722	4	,	,	PUNCT
ejpam-4412	722	5	faculty	faculty	NOUN
ejpam-4412	722	6	of	of	ADP
ejpam-4412	722	7	mathematics	mathematic	NOUN
ejpam-4412	722	8	,	,	PUNCT
ejpam-4412	722	9	belgrade	belgrade	PROPN
ejpam-4412	722	10	,	,	PUNCT
ejpam-4412	722	11	2014	2014	NUM
ejpam-4412	722	12	.	.	PUNCT
ejpam-4412	723	1	[	[	X
ejpam-4412	723	2	4	4	X
ejpam-4412	723	3	]	]	AUX
ejpam-4412	723	4	v.	v.	ADP
ejpam-4412	723	5	todorčevič.	todorčevič.	NOUN
ejpam-4412	723	6	harmonic	harmonic	VERB
ejpam-4412	723	7	quasiconformal	quasiconformal	ADJ
ejpam-4412	723	8	mappings	mapping	NOUN
ejpam-4412	723	9	and	and	CCONJ
ejpam-4412	723	10	hyperbolic	hyperbolic	ADJ
ejpam-4412	723	11	type	type	NOUN
ejpam-4412	723	12	metrics	metric	NOUN
ejpam-4412	723	13	.	.	PUNCT
ejpam-4412	724	1	springer	springer	NOUN
ejpam-4412	724	2	nature	nature	PROPN
ejpam-4412	724	3	,	,	PUNCT
ejpam-4412	724	4	switzerland	switzerland	PROPN
ejpam-4412	724	5	ag	ag	PROPN
ejpam-4412	724	6	,	,	PUNCT
ejpam-4412	724	7	2019	2019	NUM
ejpam-4412	724	8	.	.	PUNCT
ejpam-4412	725	1	[	[	X
ejpam-4412	725	2	5	5	X
ejpam-4412	725	3	]	]	PUNCT
ejpam-4412	725	4	v.	v.	ADP
ejpam-4412	725	5	todorčevič.	todorčevič.	NOUN
ejpam-4412	725	6	subharmonic	subharmonic	ADJ
ejpam-4412	725	7	behavior	behavior	NOUN
ejpam-4412	725	8	and	and	CCONJ
ejpam-4412	725	9	quasiconformal	quasiconformal	ADJ
ejpam-4412	725	10	mappings	mapping	NOUN
ejpam-4412	725	11	.	.	PUNCT
ejpam-4412	726	1	anal	anal	PROPN
ejpam-4412	726	2	.	.	PUNCT
ejpam-4412	726	3	math	math	NOUN
ejpam-4412	726	4	.	.	PUNCT
ejpam-4412	727	1	phys	phy	NOUN
ejpam-4412	727	2	.	.	PUNCT
ejpam-4412	727	3	,	,	PUNCT
ejpam-4412	727	4	9:1211–1225	9:1211–1225	NUM
ejpam-4412	727	5	,	,	PUNCT
ejpam-4412	727	6	2019	2019	NUM
ejpam-4412	727	7	.	.	PUNCT
ejpam-4412	728	1	[	[	X
ejpam-4412	728	2	6	6	NUM
ejpam-4412	728	3	]	]	PUNCT
ejpam-4412	728	4	m.	m.	NOUN
ejpam-4412	728	5	chō	chō	NOUN
ejpam-4412	728	6	and	and	CCONJ
ejpam-4412	728	7	t.	t.	NOUN
ejpam-4412	728	8	huruya	huruya	NOUN
ejpam-4412	728	9	.	.	PUNCT
ejpam-4412	729	1	p	p	X
ejpam-4412	729	2	-	-	PUNCT
ejpam-4412	729	3	hyponormal	hyponormal	ADJ
ejpam-4412	729	4	operator	operator	NOUN
ejpam-4412	729	5	for	for	ADP
ejpam-4412	729	6	0	0	NUM
ejpam-4412	729	7	<	<	X
ejpam-4412	729	8	p	p	X
ejpam-4412	729	9	<	<	X
ejpam-4412	729	10	1	1	NUM
ejpam-4412	729	11	2	2	NUM
ejpam-4412	729	12	.	.	PUNCT
ejpam-4412	730	1	commentations	commentation	NOUN
ejpam-4412	730	2	mthematicae	mthematicae	PROPN
ejpam-4412	730	3	,	,	PUNCT
ejpam-4412	730	4	33:23–29	33:23–29	NUM
ejpam-4412	730	5	,	,	PUNCT
ejpam-4412	730	6	1993	1993	NUM
ejpam-4412	730	7	.	.	PUNCT
ejpam-4412	731	1	references	reference	NOUN
ejpam-4412	731	2	1088	1088	NUM
ejpam-4412	731	3	[	[	X
ejpam-4412	731	4	7	7	NUM
ejpam-4412	731	5	]	]	X
ejpam-4412	731	6	m.	m.	NOUN
ejpam-4412	731	7	chō	chō	NOUN
ejpam-4412	731	8	and	and	CCONJ
ejpam-4412	731	9	k.	k.	PROPN
ejpam-4412	731	10	tanahashi	tanahashi	PROPN
ejpam-4412	731	11	.	.	PUNCT
ejpam-4412	732	1	isolated	isolated	ADJ
ejpam-4412	732	2	point	point	NOUN
ejpam-4412	732	3	of	of	ADP
ejpam-4412	732	4	spectrum	spectrum	NOUN
ejpam-4412	732	5	of	of	ADP
ejpam-4412	732	6	p	p	NOUN
ejpam-4412	732	7	-	-	PUNCT
ejpam-4412	732	8	hyponormal	hyponormal	ADJ
ejpam-4412	732	9	,	,	PUNCT
ejpam-4412	732	10	loghyponormal	loghyponormal	ADJ
ejpam-4412	732	11	operators	operator	NOUN
ejpam-4412	732	12	.	.	PUNCT
ejpam-4412	733	1	integral	integral	ADJ
ejpam-4412	733	2	equations	equation	NOUN
ejpam-4412	733	3	operator	operator	NOUN
ejpam-4412	733	4	theory	theory	NOUN
ejpam-4412	733	5	,	,	PUNCT
ejpam-4412	733	6	43:379–384	43:379–384	PROPN
ejpam-4412	733	7	,	,	PUNCT
ejpam-4412	733	8	2002	2002	NUM
ejpam-4412	733	9	.	.	PUNCT
ejpam-4412	734	1	[	[	X
ejpam-4412	734	2	8	8	NUM
ejpam-4412	734	3	]	]	X
ejpam-4412	734	4	m.	m.	NOUN
ejpam-4412	734	5	chō	chō	NOUN
ejpam-4412	734	6	and	and	CCONJ
ejpam-4412	734	7	k.	k.	PROPN
ejpam-4412	734	8	yamazaki	yamazaki	PROPN
ejpam-4412	734	9	.	.	PUNCT
ejpam-4412	735	1	an	an	DET
ejpam-4412	735	2	operator	operator	NOUN
ejpam-4412	735	3	transform	transform	VERB
ejpam-4412	735	4	from	from	ADP
ejpam-4412	735	5	class	class	NOUN
ejpam-4412	735	6	a	a	PRON
ejpam-4412	735	7	to	to	ADP
ejpam-4412	735	8	the	the	DET
ejpam-4412	735	9	class	class	NOUN
ejpam-4412	735	10	of	of	ADP
ejpam-4412	735	11	hyponormal	hyponormal	ADJ
ejpam-4412	735	12	operators	operator	NOUN
ejpam-4412	735	13	and	and	CCONJ
ejpam-4412	735	14	its	its	PRON
ejpam-4412	735	15	application	application	NOUN
ejpam-4412	735	16	.	.	PUNCT
ejpam-4412	736	1	integral	integral	ADJ
ejpam-4412	736	2	equations	equation	NOUN
ejpam-4412	736	3	operator	operator	NOUN
ejpam-4412	736	4	theory	theory	NOUN
ejpam-4412	736	5	,	,	PUNCT
ejpam-4412	736	6	53:497–508	53:497–508	PROPN
ejpam-4412	736	7	,	,	PUNCT
ejpam-4412	736	8	2005	2005	NUM
ejpam-4412	736	9	.	.	PUNCT
ejpam-4412	737	1	[	[	X
ejpam-4412	737	2	9	9	NUM
ejpam-4412	737	3	]	]	PUNCT
ejpam-4412	737	4	c.r.putnam	c.r.putnam	NOUN
ejpam-4412	737	5	.	.	PUNCT
ejpam-4412	738	1	ranges	range	NOUN
ejpam-4412	738	2	of	of	ADP
ejpam-4412	738	3	normal	normal	ADJ
ejpam-4412	738	4	and	and	CCONJ
ejpam-4412	738	5	subnormal	subnormal	ADJ
ejpam-4412	738	6	operators	operator	NOUN
ejpam-4412	738	7	.	.	PUNCT
ejpam-4412	739	1	michigan	michigan	PROPN
ejpam-4412	739	2	math	math	PROPN
ejpam-4412	739	3	.	.	PUNCT
ejpam-4412	740	1	j.	j.	PROPN
ejpam-4412	740	2	,	,	PUNCT
ejpam-4412	740	3	18:33	18:33	NUM
ejpam-4412	740	4	–	–	PUNCT
ejpam-4412	740	5	36	36	NUM
ejpam-4412	740	6	,	,	PUNCT
ejpam-4412	740	7	1971	1971	NUM
ejpam-4412	740	8	.	.	PUNCT
ejpam-4412	741	1	[	[	X
ejpam-4412	741	2	10	10	NUM
ejpam-4412	741	3	]	]	PUNCT
ejpam-4412	741	4	c.r.putnam	c.r.putnam	PROPN
ejpam-4412	741	5	.	.	PUNCT
ejpam-4412	741	6	hyponormal	hyponormal	ADJ
ejpam-4412	741	7	contractions	contraction	NOUN
ejpam-4412	741	8	and	and	CCONJ
ejpam-4412	741	9	strong	strong	ADJ
ejpam-4412	741	10	power	power	NOUN
ejpam-4412	741	11	convergence	convergence	NOUN
ejpam-4412	741	12	.	.	PUNCT
ejpam-4412	742	1	integral	integral	ADJ
ejpam-4412	742	2	equations	equation	NOUN
ejpam-4412	742	3	operator	operator	NOUN
ejpam-4412	742	4	theory	theory	NOUN
ejpam-4412	742	5	,	,	PUNCT
ejpam-4412	742	6	57:531–538	57:531–538	NUM
ejpam-4412	742	7	,	,	PUNCT
ejpam-4412	742	8	1975	1975	NUM
ejpam-4412	742	9	.	.	PUNCT
ejpam-4412	743	1	[	[	X
ejpam-4412	743	2	11	11	NUM
ejpam-4412	743	3	]	]	PUNCT
ejpam-4412	743	4	r.	r.	PROPN
ejpam-4412	743	5	g.	g.	PROPN
ejpam-4412	743	6	douglas	douglas	PROPN
ejpam-4412	743	7	.	.	PUNCT
ejpam-4412	744	1	on	on	ADP
ejpam-4412	744	2	majorization	majorization	NOUN
ejpam-4412	744	3	,	,	PUNCT
ejpam-4412	744	4	factorization	factorization	NOUN
ejpam-4412	744	5	,	,	PUNCT
ejpam-4412	744	6	and	and	CCONJ
ejpam-4412	744	7	range	range	NOUN
ejpam-4412	744	8	inclusion	inclusion	NOUN
ejpam-4412	744	9	of	of	ADP
ejpam-4412	744	10	operators	operator	NOUN
ejpam-4412	744	11	on	on	ADP
ejpam-4412	744	12	hilbert	hilbert	NOUN
ejpam-4412	744	13	space	space	NOUN
ejpam-4412	744	14	.	.	PUNCT
ejpam-4412	745	1	proc	proc	PROPN
ejpam-4412	745	2	.	.	PUNCT
ejpam-4412	746	1	amer	amer	PROPN
ejpam-4412	746	2	.	.	PUNCT
ejpam-4412	746	3	math	math	PROPN
ejpam-4412	746	4	.	.	PUNCT
ejpam-4412	747	1	soc	soc	PROPN
ejpam-4412	747	2	.	.	PUNCT
ejpam-4412	747	3	,	,	PUNCT
ejpam-4412	747	4	17:413–415	17:413–415	NUM
ejpam-4412	747	5	,	,	PUNCT
ejpam-4412	747	6	1966	1966	NUM
ejpam-4412	747	7	.	.	PUNCT
ejpam-4412	748	1	[	[	X
ejpam-4412	748	2	12	12	NUM
ejpam-4412	748	3	]	]	PUNCT
ejpam-4412	748	4	b.	b.	PROPN
ejpam-4412	748	5	p.	p.	PROPN
ejpam-4412	748	6	duggal	duggal	PROPN
ejpam-4412	748	7	.	.	PUNCT
ejpam-4412	749	1	quasi	quasi	ADJ
ejpam-4412	749	2	-	-	ADJ
ejpam-4412	749	3	similar	similar	ADJ
ejpam-4412	749	4	p	p	ADJ
ejpam-4412	749	5	-	-	PUNCT
ejpam-4412	749	6	hyponormal	hyponormal	ADJ
ejpam-4412	749	7	operators	operator	NOUN
ejpam-4412	749	8	.	.	PUNCT
ejpam-4412	750	1	integral	integral	ADJ
ejpam-4412	750	2	equations	equation	NOUN
ejpam-4412	750	3	operator	operator	NOUN
ejpam-4412	750	4	theory	theory	NOUN
ejpam-4412	750	5	,	,	PUNCT
ejpam-4412	750	6	26:338–345	26:338–345	PROPN
ejpam-4412	750	7	,	,	PUNCT
ejpam-4412	750	8	1996	1996	NUM
ejpam-4412	750	9	.	.	PUNCT
ejpam-4412	751	1	[	[	X
ejpam-4412	751	2	13	13	NUM
ejpam-4412	751	3	]	]	PUNCT
ejpam-4412	751	4	m.	m.	NOUN
ejpam-4412	751	5	fujii	fujii	PROPN
ejpam-4412	751	6	,	,	PUNCT
ejpam-4412	751	7	d.	d.	PROPN
ejpam-4412	751	8	jung	jung	PROPN
ejpam-4412	751	9	,	,	PUNCT
ejpam-4412	751	10	s.	s.	PROPN
ejpam-4412	751	11	h.	h.	PROPN
ejpam-4412	751	12	lee	lee	PROPN
ejpam-4412	751	13	.	.	PROPN
ejpam-4412	751	14	,	,	PUNCT
ejpam-4412	751	15	m.	m.	PROPN
ejpam-4412	751	16	y.	y.	PROPN
ejpam-4412	751	17	lee	lee	PROPN
ejpam-4412	751	18	.	.	PROPN
ejpam-4412	751	19	,	,	PUNCT
ejpam-4412	751	20	and	and	CCONJ
ejpam-4412	751	21	r.	r.	PROPN
ejpam-4412	751	22	nakamoto	nakamoto	NOUN
ejpam-4412	751	23	.	.	PUNCT
ejpam-4412	752	1	some	some	DET
ejpam-4412	752	2	classes	class	NOUN
ejpam-4412	752	3	of	of	ADP
ejpam-4412	752	4	operators	operator	NOUN
ejpam-4412	752	5	related	relate	VERB
ejpam-4412	752	6	to	to	ADP
ejpam-4412	752	7	paranormal	paranormal	NOUN
ejpam-4412	752	8	and	and	CCONJ
ejpam-4412	752	9	log	log	NOUN
ejpam-4412	752	10	-	-	PUNCT
ejpam-4412	752	11	hyponormal	hyponormal	ADJ
ejpam-4412	752	12	operators	operator	NOUN
ejpam-4412	752	13	.	.	PUNCT
ejpam-4412	753	1	math	math	NOUN
ejpam-4412	753	2	.	.	PUNCT
ejpam-4412	754	1	japon	japon	PROPN
ejpam-4412	754	2	.	.	PROPN
ejpam-4412	754	3	,	,	PUNCT
ejpam-4412	754	4	51:395–402	51:395–402	PROPN
ejpam-4412	754	5	,	,	PUNCT
ejpam-4412	754	6	2000	2000	NUM
ejpam-4412	754	7	.	.	PUNCT
ejpam-4412	755	1	[	[	X
ejpam-4412	755	2	14	14	NUM
ejpam-4412	755	3	]	]	PUNCT
ejpam-4412	755	4	t.	t.	PROPN
ejpam-4412	755	5	furuta	furuta	PROPN
ejpam-4412	755	6	.	.	PUNCT
ejpam-4412	756	1	a	a	DET
ejpam-4412	756	2	≥	≥	NOUN
ejpam-4412	756	3	b	b	NOUN
ejpam-4412	756	4	≥	≥	NUM
ejpam-4412	756	5	o	o	NOUN
ejpam-4412	756	6	assures	assures	PROPN
ejpam-4412	756	7	(	(	PUNCT
ejpam-4412	756	8	brapbr	brapbr	PROPN
ejpam-4412	756	9	)	)	PUNCT
ejpam-4412	756	10	1	1	NUM
ejpam-4412	756	11	q	q	NOUN
ejpam-4412	756	12	≥	≥	NOUN
ejpam-4412	756	13	b	b	X
ejpam-4412	756	14	p+2r	p+2r	NOUN
ejpam-4412	756	15	q	q	NOUN
ejpam-4412	756	16	for	for	ADP
ejpam-4412	756	17	r	r	NOUN
ejpam-4412	756	18	≥	≥	NOUN
ejpam-4412	756	19	0	0	NUM
ejpam-4412	756	20	,	,	PUNCT
ejpam-4412	756	21	p	p	PRON
ejpam-4412	756	22	≥	≥	NOUN
ejpam-4412	756	23	0	0	NUM
ejpam-4412	756	24	,	,	PUNCT
ejpam-4412	756	25	q	q	X
ejpam-4412	756	26	≥	≥	NOUN
ejpam-4412	756	27	1	1	NUM
ejpam-4412	756	28	with	with	ADP
ejpam-4412	756	29	(	(	PUNCT
ejpam-4412	756	30	1	1	NUM
ejpam-4412	756	31	+	+	NUM
ejpam-4412	756	32	2r)q	2r)q	NUM
ejpam-4412	756	33	≥	≥	NUM
ejpam-4412	756	34	(	(	PUNCT
ejpam-4412	756	35	p+	p+	NOUN
ejpam-4412	756	36	2r	2r	NUM
ejpam-4412	756	37	)	)	PUNCT
ejpam-4412	756	38	.	.	PUNCT
ejpam-4412	757	1	proc	proc	PROPN
ejpam-4412	757	2	.	.	PUNCT
ejpam-4412	758	1	amer	amer	PROPN
ejpam-4412	758	2	.	.	PUNCT
ejpam-4412	758	3	math	math	PROPN
ejpam-4412	758	4	.	.	PUNCT
ejpam-4412	759	1	soc	soc	PROPN
ejpam-4412	759	2	.	.	PUNCT
ejpam-4412	759	3	,	,	PUNCT
ejpam-4412	759	4	101:85–88	101:85–88	NUM
ejpam-4412	759	5	,	,	PUNCT
ejpam-4412	759	6	1987	1987	NUM
ejpam-4412	759	7	.	.	PUNCT
ejpam-4412	760	1	[	[	X
ejpam-4412	760	2	15	15	NUM
ejpam-4412	760	3	]	]	X
ejpam-4412	760	4	li	li	PROPN
ejpam-4412	760	5	haiying	haiying	PROPN
ejpam-4412	760	6	.	.	PUNCT
ejpam-4412	761	1	powers	power	NOUN
ejpam-4412	761	2	of	of	ADP
ejpam-4412	761	3	an	an	DET
ejpam-4412	761	4	invertible	invertible	ADJ
ejpam-4412	761	5	(	(	PUNCT
ejpam-4412	761	6	s	s	PROPN
ejpam-4412	761	7	,	,	PUNCT
ejpam-4412	761	8	p)-w	p)-w	PROPN
ejpam-4412	761	9	-	-	PUNCT
ejpam-4412	761	10	hyponormal	hyponormal	ADJ
ejpam-4412	761	11	operator	operator	NOUN
ejpam-4412	761	12	.	.	PUNCT
ejpam-4412	762	1	acta	acta	PROPN
ejpam-4412	762	2	math	math	PROPN
ejpam-4412	762	3	.	.	PUNCT
ejpam-4412	763	1	scientia	scientia	PROPN
ejpam-4412	763	2	,	,	PUNCT
ejpam-4412	763	3	28b(2):282–288	28b(2):282–288	NUM
ejpam-4412	763	4	,	,	PUNCT
ejpam-4412	763	5	2008	2008	NUM
ejpam-4412	763	6	.	.	PUNCT
ejpam-4412	764	1	[	[	X
ejpam-4412	764	2	16	16	NUM
ejpam-4412	764	3	]	]	X
ejpam-4412	764	4	f.	f.	PROPN
ejpam-4412	764	5	hansen	hansen	PROPN
ejpam-4412	764	6	.	.	PUNCT
ejpam-4412	765	1	an	an	DET
ejpam-4412	765	2	equality	equality	NOUN
ejpam-4412	765	3	.	.	PUNCT
ejpam-4412	766	1	math	math	NOUN
ejpam-4412	766	2	.	.	PUNCT
ejpam-4412	767	1	ann	ann	PROPN
ejpam-4412	767	2	.	.	PROPN
ejpam-4412	767	3	,	,	PUNCT
ejpam-4412	767	4	246:249––250	246:249––250	NUM
ejpam-4412	767	5	,	,	PUNCT
ejpam-4412	767	6	1980	1980	NUM
ejpam-4412	767	7	.	.	PUNCT
ejpam-4412	768	1	[	[	X
ejpam-4412	768	2	17	17	NUM
ejpam-4412	768	3	]	]	X
ejpam-4412	768	4	f.	f.	PROPN
ejpam-4412	768	5	hansen	hansen	PROPN
ejpam-4412	768	6	and	and	CCONJ
ejpam-4412	768	7	g.	g.	PROPN
ejpam-4412	768	8	k.	k.	PROPN
ejpam-4412	768	9	pedersen	pedersen	PROPN
ejpam-4412	768	10	.	.	PUNCT
ejpam-4412	769	1	jensen	jensen	PROPN
ejpam-4412	769	2	’s	’s	PART
ejpam-4412	769	3	inequality	inequality	NOUN
ejpam-4412	769	4	for	for	ADP
ejpam-4412	769	5	operators	operator	NOUN
ejpam-4412	769	6	and	and	CCONJ
ejpam-4412	769	7	l	l	PROPN
ejpam-4412	769	8	owner	owner	NOUN
ejpam-4412	769	9	’s	’s	PART
ejpam-4412	769	10	theorem	theorem	PROPN
ejpam-4412	769	11	.	.	PROPN
ejpam-4412	769	12	math	math	PROPN
ejpam-4412	769	13	.	.	PUNCT
ejpam-4412	770	1	ann	ann	PROPN
ejpam-4412	770	2	.	.	PROPN
ejpam-4412	770	3	,	,	PUNCT
ejpam-4412	770	4	258:229–241	258:229–241	NUM
ejpam-4412	770	5	,	,	PUNCT
ejpam-4412	770	6	1982	1982	NUM
ejpam-4412	770	7	.	.	PUNCT
ejpam-4412	771	1	[	[	X
ejpam-4412	771	2	18	18	NUM
ejpam-4412	771	3	]	]	PUNCT
ejpam-4412	771	4	m.	m.	PROPN
ejpam-4412	771	5	ito	ito	PROPN
ejpam-4412	771	6	.	.	PUNCT
ejpam-4412	772	1	some	some	DET
ejpam-4412	772	2	classes	class	NOUN
ejpam-4412	772	3	of	of	ADP
ejpam-4412	772	4	operators	operator	NOUN
ejpam-4412	772	5	with	with	ADP
ejpam-4412	772	6	generalised	generalise	VERB
ejpam-4412	772	7	aluthege	aluthege	NOUN
ejpam-4412	772	8	transformations	transformation	NOUN
ejpam-4412	772	9	.	.	PUNCT
ejpam-4412	773	1	sut	sut	PROPN
ejpam-4412	773	2	j.	j.	PROPN
ejpam-4412	773	3	math	math	PROPN
ejpam-4412	773	4	.	.	PUNCT
ejpam-4412	773	5	,	,	PUNCT
ejpam-4412	773	6	35:149––165	35:149––165	NUM
ejpam-4412	773	7	,	,	PUNCT
ejpam-4412	773	8	1999	1999	NUM
ejpam-4412	773	9	.	.	PUNCT
ejpam-4412	774	1	[	[	X
ejpam-4412	774	2	19	19	NUM
ejpam-4412	774	3	]	]	PUNCT
ejpam-4412	774	4	m.	m.	NOUN
ejpam-4412	774	5	ito	ito	PROPN
ejpam-4412	774	6	and	and	CCONJ
ejpam-4412	774	7	t.	t.	PROPN
ejpam-4412	774	8	yamazaki	yamazaki	PROPN
ejpam-4412	774	9	.	.	PUNCT
ejpam-4412	775	1	relations	relation	NOUN
ejpam-4412	775	2	between	between	ADP
ejpam-4412	775	3	two	two	NUM
ejpam-4412	775	4	inequalities	inequality	NOUN
ejpam-4412	775	5	(	(	PUNCT
ejpam-4412	775	6	b	b	NOUN
ejpam-4412	775	7	r	r	NOUN
ejpam-4412	775	8	2apb	2apb	NUM
ejpam-4412	775	9	r	r	NOUN
ejpam-4412	775	10	2	2	NUM
ejpam-4412	775	11	)	)	PUNCT
ejpam-4412	775	12	r	r	NOUN
ejpam-4412	775	13	p+r	p+r	NUM
ejpam-4412	775	14	≥	≥	NOUN
ejpam-4412	775	15	br	br	NOUN
ejpam-4412	775	16	and	and	CCONJ
ejpam-4412	775	17	ap	ap	PROPN
ejpam-4412	775	18	≥	≥	PROPN
ejpam-4412	775	19	(	(	PUNCT
ejpam-4412	775	20	a	a	DET
ejpam-4412	775	21	p	p	X
ejpam-4412	775	22	2bra	2bra	PROPN
ejpam-4412	775	23	p	p	NOUN
ejpam-4412	775	24	2	2	NUM
ejpam-4412	775	25	)	)	PUNCT
ejpam-4412	775	26	r	r	NOUN
ejpam-4412	775	27	p+r	p+r	NUM
ejpam-4412	775	28	and	and	CCONJ
ejpam-4412	775	29	their	their	PRON
ejpam-4412	775	30	applications	application	NOUN
ejpam-4412	775	31	.	.	PUNCT
ejpam-4412	776	1	integral	integral	ADJ
ejpam-4412	776	2	equations	equation	NOUN
ejpam-4412	776	3	operator	operator	NOUN
ejpam-4412	776	4	theory	theory	NOUN
ejpam-4412	776	5	,	,	PUNCT
ejpam-4412	776	6	44:442–450	44:442–450	PROPN
ejpam-4412	776	7	,	,	PUNCT
ejpam-4412	776	8	2002	2002	NUM
ejpam-4412	776	9	.	.	PUNCT
ejpam-4412	777	1	[	[	X
ejpam-4412	777	2	20	20	NUM
ejpam-4412	777	3	]	]	PUNCT
ejpam-4412	777	4	i.	i.	PROPN
ejpam-4412	777	5	h.	h.	PROPN
ejpam-4412	777	6	jeon	jeon	PROPN
ejpam-4412	777	7	and	and	CCONJ
ejpam-4412	777	8	b.	b.	PROPN
ejpam-4412	777	9	p.	p.	PROPN
ejpam-4412	777	10	duggal	duggal	PROPN
ejpam-4412	777	11	.	.	PUNCT
ejpam-4412	778	1	p	p	X
ejpam-4412	778	2	-	-	PUNCT
ejpam-4412	778	3	hyponormal	hyponormal	ADJ
ejpam-4412	778	4	operators	operator	NOUN
ejpam-4412	778	5	and	and	CCONJ
ejpam-4412	778	6	quasisimilarity	quasisimilarity	NOUN
ejpam-4412	778	7	.	.	PUNCT
ejpam-4412	779	1	integral	integral	ADJ
ejpam-4412	779	2	equations	equation	NOUN
ejpam-4412	779	3	operator	operator	NOUN
ejpam-4412	779	4	theory	theory	NOUN
ejpam-4412	779	5	,	,	PUNCT
ejpam-4412	779	6	49(3):397––403	49(3):397––403	NOUN
ejpam-4412	779	7	,	,	PUNCT
ejpam-4412	779	8	2004	2004	NUM
ejpam-4412	779	9	.	.	PUNCT
ejpam-4412	780	1	[	[	X
ejpam-4412	780	2	21	21	NUM
ejpam-4412	780	3	]	]	PUNCT
ejpam-4412	780	4	m.	m.	NOUN
ejpam-4412	780	5	chō	chō	NOUN
ejpam-4412	780	6	,	,	PUNCT
ejpam-4412	780	7	m.h.m	m.h.m	NOUN
ejpam-4412	780	8	rashid	rashid	PROPN
ejpam-4412	780	9	,	,	PUNCT
ejpam-4412	780	10	k.	k.	PROPN
ejpam-4412	780	11	tanahashi	tanahashi	PROPN
ejpam-4412	780	12	and	and	CCONJ
ejpam-4412	780	13	a.	a.	NOUN
ejpam-4412	780	14	uchiyama	uchiyama	NOUN
ejpam-4412	780	15	.	.	PUNCT
ejpam-4412	781	1	spectrum	spectrum	NOUN
ejpam-4412	781	2	of	of	ADP
ejpam-4412	781	3	class	class	NOUN
ejpam-4412	781	4	pwa(s	pwa(s	PROPN
ejpam-4412	781	5	,	,	PUNCT
ejpam-4412	781	6	t	t	PROPN
ejpam-4412	781	7	)	)	PUNCT
ejpam-4412	781	8	operators	operator	NOUN
ejpam-4412	781	9	.	.	PUNCT
ejpam-4412	782	1	acta	acta	PROPN
ejpam-4412	782	2	sci	sci	PROPN
ejpam-4412	782	3	.	.	PROPN
ejpam-4412	782	4	math	math	PROPN
ejpam-4412	782	5	.	.	PUNCT
ejpam-4412	783	1	(	(	PUNCT
ejpam-4412	783	2	szeged	szeged	PROPN
ejpam-4412	783	3	)	)	PUNCT
ejpam-4412	783	4	,	,	PUNCT
ejpam-4412	783	5	82(3	82(3	NUM
ejpam-4412	783	6	-	-	SYM
ejpam-4412	783	7	4):641–649	4):641–649	NOUN
ejpam-4412	783	8	,	,	PUNCT
ejpam-4412	783	9	2016	2016	NUM
ejpam-4412	783	10	.	.	PUNCT
ejpam-4412	784	1	[	[	X
ejpam-4412	784	2	22	22	NUM
ejpam-4412	784	3	]	]	X
ejpam-4412	784	4	m.h.m.rashid	m.h.m.rashid	NOUN
ejpam-4412	784	5	.	.	PUNCT
ejpam-4412	785	1	an	an	DET
ejpam-4412	785	2	extension	extension	NOUN
ejpam-4412	785	3	of	of	ADP
ejpam-4412	785	4	fuglede	fuglede	NOUN
ejpam-4412	785	5	-	-	PUNCT
ejpam-4412	785	6	putnam	putnam	NOUN
ejpam-4412	785	7	theorem	theorem	NOUN
ejpam-4412	785	8	for	for	ADP
ejpam-4412	785	9	w	w	NOUN
ejpam-4412	785	10	-	-	PUNCT
ejpam-4412	785	11	hyponormal	hyponormal	ADJ
ejpam-4412	785	12	operators	operator	NOUN
ejpam-4412	785	13	.	.	PUNCT
ejpam-4412	786	1	afr	afr	PROPN
ejpam-4412	786	2	.	.	PUNCT
ejpam-4412	787	1	diaspora	diaspora	PROPN
ejpam-4412	787	2	j.	j.	PROPN
ejpam-4412	787	3	math	math	PROPN
ejpam-4412	787	4	.	.	PUNCT
ejpam-4412	788	1	(	(	PUNCT
ejpam-4412	788	2	n.s	n.s	PROPN
ejpam-4412	788	3	.	.	PROPN
ejpam-4412	788	4	)	)	PUNCT
ejpam-4412	788	5	,	,	PUNCT
ejpam-4412	788	6	14(1):106––118	14(1):106––118	NUM
ejpam-4412	788	7	,	,	PUNCT
ejpam-4412	788	8	2012	2012	NUM
ejpam-4412	788	9	.	.	PUNCT
ejpam-4412	789	1	references	reference	NOUN
ejpam-4412	789	2	1089	1089	NUM
ejpam-4412	789	3	[	[	X
ejpam-4412	789	4	23	23	NUM
ejpam-4412	789	5	]	]	X
ejpam-4412	789	6	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	789	7	.	.	PUNCT
ejpam-4412	789	8	class	class	NOUN
ejpam-4412	789	9	wa(s	wa(s	PROPN
ejpam-4412	789	10	,	,	PUNCT
ejpam-4412	789	11	t	t	NOUN
ejpam-4412	789	12	)	)	PUNCT
ejpam-4412	789	13	operators	operator	NOUN
ejpam-4412	789	14	and	and	CCONJ
ejpam-4412	789	15	quasisimilarity	quasisimilarity	NOUN
ejpam-4412	789	16	.	.	PUNCT
ejpam-4412	790	1	port	port	NOUN
ejpam-4412	790	2	.	.	PUNCT
ejpam-4412	791	1	math	math	NOUN
ejpam-4412	791	2	.	.	PUNCT
ejpam-4412	791	3	,	,	PUNCT
ejpam-4412	792	1	69(4):305	69(4):305	ADJ
ejpam-4412	792	2	–	–	PUNCT
ejpam-4412	792	3	320	320	NUM
ejpam-4412	792	4	,	,	PUNCT
ejpam-4412	792	5	2012	2012	NUM
ejpam-4412	792	6	.	.	PUNCT
ejpam-4412	793	1	[	[	X
ejpam-4412	793	2	24	24	NUM
ejpam-4412	793	3	]	]	X
ejpam-4412	793	4	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	793	5	.	.	PUNCT
ejpam-4412	793	6	fuglede	fuglede	PROPN
ejpam-4412	793	7	-	-	PUNCT
ejpam-4412	793	8	putnam	putnam	PROPN
ejpam-4412	793	9	type	type	NOUN
ejpam-4412	793	10	theorems	theorem	NOUN
ejpam-4412	793	11	via	via	ADP
ejpam-4412	793	12	the	the	DET
ejpam-4412	793	13	generalized	generalized	ADJ
ejpam-4412	793	14	aluthge	aluthge	ADJ
ejpam-4412	793	15	transform	transform	NOUN
ejpam-4412	793	16	.	.	PUNCT
ejpam-4412	794	1	revista	revista	PROPN
ejpam-4412	794	2	de	de	X
ejpam-4412	794	3	la	la	PROPN
ejpam-4412	794	4	real	real	PROPN
ejpam-4412	794	5	academia	academia	PROPN
ejpam-4412	794	6	de	de	PROPN
ejpam-4412	794	7	ciencias	ciencias	PROPN
ejpam-4412	794	8	exactas	exacta	NOUN
ejpam-4412	794	9	,	,	PUNCT
ejpam-4412	794	10	fisicas	fisicas	PROPN
ejpam-4412	794	11	y	y	PROPN
ejpam-4412	794	12	naturales	naturales	PROPN
ejpam-4412	794	13	serie	serie	VERB
ejpam-4412	794	14	a	a	DET
ejpam-4412	794	15	:	:	SYM
ejpam-4412	794	16	matematicas	matematicas	PROPN
ejpam-4412	794	17	,	,	PUNCT
ejpam-4412	794	18	108(2):1021––1034	108(2):1021––1034	NUM
ejpam-4412	794	19	,	,	PUNCT
ejpam-4412	794	20	2014	2014	NUM
ejpam-4412	794	21	.	.	PUNCT
ejpam-4412	795	1	[	[	X
ejpam-4412	795	2	25	25	NUM
ejpam-4412	795	3	]	]	X
ejpam-4412	795	4	m.h.m.rashid	m.h.m.rashid	PROPN
ejpam-4412	795	5	.	.	PUNCT
ejpam-4412	795	6	a	a	DET
ejpam-4412	795	7	note	note	NOUN
ejpam-4412	795	8	on	on	ADP
ejpam-4412	795	9	class	class	NOUN
ejpam-4412	795	10	p	p	NOUN
ejpam-4412	795	11	-	-	PUNCT
ejpam-4412	795	12	wa(s	wa(s	NUM
ejpam-4412	795	13	,	,	PUNCT
ejpam-4412	795	14	t	t	NOUN
ejpam-4412	795	15	)	)	PUNCT
ejpam-4412	795	16	operators	operator	NOUN
ejpam-4412	795	17	.	.	PUNCT
ejpam-4412	796	1	filomat	filomat	NOUN
ejpam-4412	796	2	,	,	PUNCT
ejpam-4412	796	3	36(5):1675––1684	36(5):1675––1684	NUM
ejpam-4412	796	4	,	,	PUNCT
ejpam-4412	796	5	2022	2022	NUM
ejpam-4412	796	6	.	.	PUNCT
ejpam-4412	797	1	[	[	X
ejpam-4412	797	2	26	26	NUM
ejpam-4412	797	3	]	]	PUNCT
ejpam-4412	797	4	t.	t.	NOUN
ejpam-4412	797	5	prasad	prasad	PROPN
ejpam-4412	797	6	and	and	CCONJ
ejpam-4412	797	7	k.	k.	PROPN
ejpam-4412	797	8	tanahashi	tanahashi	PROPN
ejpam-4412	797	9	.	.	PUNCT
ejpam-4412	798	1	on	on	ADP
ejpam-4412	798	2	class	class	NOUN
ejpam-4412	798	3	p	p	NOUN
ejpam-4412	798	4	-	-	PUNCT
ejpam-4412	798	5	wa(s	wa(s	NUM
ejpam-4412	798	6	,	,	PUNCT
ejpam-4412	798	7	t	t	NOUN
ejpam-4412	798	8	)	)	PUNCT
ejpam-4412	798	9	operators	operator	NOUN
ejpam-4412	798	10	.	.	PUNCT
ejpam-4412	799	1	functional	functional	ADJ
ejpam-4412	799	2	analysis	analysis	NOUN
ejpam-4412	799	3	,	,	PUNCT
ejpam-4412	799	4	approximation	approximation	NOUN
ejpam-4412	799	5	computation	computation	NOUN
ejpam-4412	799	6	,	,	PUNCT
ejpam-4412	799	7	6(2):39–42	6(2):39–42	NUM
ejpam-4412	799	8	,	,	PUNCT
ejpam-4412	799	9	2014	2014	NUM
ejpam-4412	799	10	.	.	PUNCT
ejpam-4412	800	1	[	[	X
ejpam-4412	800	2	27	27	NUM
ejpam-4412	800	3	]	]	X
ejpam-4412	800	4	s.	s.	PROPN
ejpam-4412	800	5	m.	m.	PROPN
ejpam-4412	800	6	patel	patel	PROPN
ejpam-4412	800	7	,	,	PUNCT
ejpam-4412	800	8	k.	k.	PROPN
ejpam-4412	800	9	tanahashi	tanahashi	PROPN
ejpam-4412	800	10	,	,	PUNCT
ejpam-4412	800	11	a.	a.	NOUN
ejpam-4412	800	12	uchiyama	uchiyama	NOUN
ejpam-4412	800	13	and	and	CCONJ
ejpam-4412	800	14	m.	m.	NOUN
ejpam-4412	800	15	yanagida	yanagida	PROPN
ejpam-4412	800	16	.	.	PUNCT
ejpam-4412	801	1	quasinormality	quasinormality	NOUN
ejpam-4412	801	2	and	and	CCONJ
ejpam-4412	801	3	fuglede	fuglede	NOUN
ejpam-4412	801	4	-	-	PUNCT
ejpam-4412	801	5	putnam	putnam	NOUN
ejpam-4412	801	6	theorem	theorem	NOUN
ejpam-4412	801	7	for	for	ADP
ejpam-4412	801	8	class	class	NOUN
ejpam-4412	801	9	a(s	a(s	PROPN
ejpam-4412	801	10	,	,	PUNCT
ejpam-4412	801	11	t	t	PROPN
ejpam-4412	801	12	)	)	PUNCT
ejpam-4412	801	13	operators	operator	NOUN
ejpam-4412	801	14	.	.	PUNCT
ejpam-4412	802	1	nihonkai	nihonkai	PROPN
ejpam-4412	802	2	math	math	PROPN
ejpam-4412	802	3	.	.	PUNCT
ejpam-4412	803	1	j.	j.	PROPN
ejpam-4412	803	2	,	,	PUNCT
ejpam-4412	803	3	17:49–67	17:49–67	NUM
ejpam-4412	803	4	,	,	PUNCT
ejpam-4412	803	5	2006	2006	NUM
ejpam-4412	803	6	.	.	PUNCT
ejpam-4412	804	1	[	[	X
ejpam-4412	804	2	28	28	NUM
ejpam-4412	804	3	]	]	X
ejpam-4412	804	4	j.	j.	PROPN
ejpam-4412	804	5	g.	g.	PROPN
ejpam-4412	804	6	stampfli	stampfli	PROPN
ejpam-4412	804	7	and	and	CCONJ
ejpam-4412	804	8	b.	b.	PROPN
ejpam-4412	804	9	l.	l.	PROPN
ejpam-4412	804	10	wadhwa	wadhwa	PROPN
ejpam-4412	804	11	.	.	PUNCT
ejpam-4412	805	1	an	an	DET
ejpam-4412	805	2	asymmetric	asymmetric	ADJ
ejpam-4412	805	3	putnam	putnam	PROPN
ejpam-4412	805	4	-	-	PUNCT
ejpam-4412	805	5	fuglede	fuglede	PROPN
ejpam-4412	805	6	theorem	theorem	NOUN
ejpam-4412	805	7	for	for	ADP
ejpam-4412	805	8	dominant	dominant	ADJ
ejpam-4412	805	9	operators	operator	NOUN
ejpam-4412	805	10	.	.	PUNCT
ejpam-4412	806	1	indiana	indiana	PROPN
ejpam-4412	806	2	univ	univ	PROPN
ejpam-4412	806	3	.	.	PUNCT
ejpam-4412	807	1	math	math	PROPN
ejpam-4412	807	2	.	.	PUNCT
ejpam-4412	807	3	,	,	PUNCT
ejpam-4412	807	4	25(4):359––365	25(4):359––365	NUM
ejpam-4412	807	5	,	,	PUNCT
ejpam-4412	807	6	1976	1976	NUM
ejpam-4412	807	7	.	.	PUNCT
ejpam-4412	808	1	[	[	X
ejpam-4412	808	2	29	29	NUM
ejpam-4412	808	3	]	]	PUNCT
ejpam-4412	808	4	k.	k.	PROPN
ejpam-4412	808	5	takahashi	takahashi	PROPN
ejpam-4412	808	6	.	.	PUNCT
ejpam-4412	809	1	on	on	ADP
ejpam-4412	809	2	the	the	DET
ejpam-4412	809	3	converse	converse	NOUN
ejpam-4412	809	4	of	of	ADP
ejpam-4412	809	5	putnam	putnam	PROPN
ejpam-4412	809	6	-	-	PUNCT
ejpam-4412	809	7	fuglede	fuglede	PROPN
ejpam-4412	809	8	theorem	theorem	PROPN
ejpam-4412	809	9	.	.	PUNCT
ejpam-4412	809	10	acta	acta	PROPN
ejpam-4412	809	11	sci	sci	PROPN
ejpam-4412	809	12	.	.	PROPN
ejpam-4412	809	13	math.(szeged	math.(szeged	NUM
ejpam-4412	809	14	)	)	PUNCT
ejpam-4412	809	15	,	,	PUNCT
ejpam-4412	809	16	43:123–125	43:123–125	PROPN
ejpam-4412	809	17	,	,	PUNCT
ejpam-4412	809	18	1981	1981	NUM
ejpam-4412	809	19	.	.	PUNCT
ejpam-4412	810	1	[	[	X
ejpam-4412	810	2	30	30	NUM
ejpam-4412	810	3	]	]	PUNCT
ejpam-4412	810	4	a.	a.	NOUN
ejpam-4412	810	5	uchiyama	uchiyama	NOUN
ejpam-4412	810	6	and	and	CCONJ
ejpam-4412	810	7	k.	k.	PROPN
ejpam-4412	810	8	tanahashi	tanahashi	PROPN
ejpam-4412	810	9	.	.	PUNCT
ejpam-4412	811	1	fuglede	fuglede	PROPN
ejpam-4412	811	2	-	-	PUNCT
ejpam-4412	811	3	putnam	putnam	PROPN
ejpam-4412	811	4	theorem	theorem	NOUN
ejpam-4412	811	5	for	for	ADP
ejpam-4412	811	6	p	p	NOUN
ejpam-4412	811	7	-	-	PUNCT
ejpam-4412	811	8	hyponormal	hyponormal	ADJ
ejpam-4412	811	9	or	or	CCONJ
ejpam-4412	811	10	loghyponormal	loghyponormal	ADJ
ejpam-4412	811	11	operators	operator	NOUN
ejpam-4412	811	12	.	.	PUNCT
ejpam-4412	812	1	glasg	glasg	PROPN
ejpam-4412	812	2	.	.	PUNCT
ejpam-4412	813	1	math	math	NOUN
ejpam-4412	813	2	.	.	PUNCT
ejpam-4412	814	1	j.	j.	PROPN
ejpam-4412	814	2	,	,	PUNCT
ejpam-4412	814	3	44:397–410	44:397–410	PROPN
ejpam-4412	814	4	,	,	PUNCT
ejpam-4412	814	5	2002	2002	NUM
ejpam-4412	814	6	.	.	PUNCT
ejpam-4412	815	1	[	[	X
ejpam-4412	815	2	31	31	NUM
ejpam-4412	815	3	]	]	PUNCT
ejpam-4412	815	4	a.	a.	NOUN
ejpam-4412	815	5	uchiyama	uchiyama	NOUN
ejpam-4412	815	6	and	and	CCONJ
ejpam-4412	815	7	k.	k.	PROPN
ejpam-4412	815	8	tanahashi	tanahashi	PROPN
ejpam-4412	815	9	.	.	PUNCT
ejpam-4412	816	1	on	on	ADP
ejpam-4412	816	2	the	the	DET
ejpam-4412	816	3	riesz	riesz	NOUN
ejpam-4412	816	4	idempotent	idempotent	NOUN
ejpam-4412	816	5	of	of	ADP
ejpam-4412	816	6	class	class	NOUN
ejpam-4412	816	7	a	a	DET
ejpam-4412	816	8	operators	operator	NOUN
ejpam-4412	816	9	.	.	PUNCT
ejpam-4412	817	1	math	math	NOUN
ejpam-4412	817	2	.	.	PUNCT
ejpam-4412	818	1	ineq	ineq	PROPN
ejpam-4412	818	2	.	.	PUNCT
ejpam-4412	819	1	appl	appl	PROPN
ejpam-4412	819	2	.	.	PROPN
ejpam-4412	819	3	,	,	PUNCT
ejpam-4412	819	4	5(2):291–298	5(2):291–298	NUM
ejpam-4412	819	5	,	,	PUNCT
ejpam-4412	819	6	2002	2002	NUM
ejpam-4412	819	7	.	.	PUNCT
ejpam-4412	820	1	[	[	X
ejpam-4412	820	2	32	32	NUM
ejpam-4412	820	3	]	]	PUNCT
ejpam-4412	820	4	l.	l.	PROPN
ejpam-4412	820	5	r.	r.	PROPN
ejpam-4412	820	6	williams	williams	PROPN
ejpam-4412	820	7	.	.	PUNCT
ejpam-4412	821	1	quasi	quasi	ADJ
ejpam-4412	821	2	-	-	NOUN
ejpam-4412	821	3	similarity	similarity	NOUN
ejpam-4412	821	4	and	and	CCONJ
ejpam-4412	821	5	hyponormal	hyponormal	ADJ
ejpam-4412	821	6	operators	operator	NOUN
ejpam-4412	821	7	.	.	PUNCT
ejpam-4412	822	1	integral	integral	ADJ
ejpam-4412	822	2	equations	equation	NOUN
ejpam-4412	822	3	operator	operator	NOUN
ejpam-4412	822	4	theory	theory	NOUN
ejpam-4412	822	5	,	,	PUNCT
ejpam-4412	822	6	5:678–686	5:678–686	NUM
ejpam-4412	822	7	,	,	PUNCT
ejpam-4412	822	8	1981	1981	NUM
ejpam-4412	822	9	.	.	PUNCT
ejpam-4412	823	1	[	[	X
ejpam-4412	823	2	33	33	NUM
ejpam-4412	823	3	]	]	PUNCT
ejpam-4412	823	4	m.	m.	NOUN
ejpam-4412	823	5	yanagida	yanagida	PROPN
ejpam-4412	823	6	.	.	PUNCT
ejpam-4412	824	1	powers	power	NOUN
ejpam-4412	824	2	of	of	ADP
ejpam-4412	824	3	class	class	NOUN
ejpam-4412	824	4	wa(s	wa(s	PROPN
ejpam-4412	824	5	,	,	PUNCT
ejpam-4412	824	6	t	t	NOUN
ejpam-4412	824	7	)	)	PUNCT
ejpam-4412	824	8	operators	operator	NOUN
ejpam-4412	824	9	with	with	ADP
ejpam-4412	824	10	generalised	generalised	ADJ
ejpam-4412	824	11	aluthge	aluthge	ADJ
ejpam-4412	824	12	transformation	transformation	NOUN
ejpam-4412	824	13	.	.	PUNCT
ejpam-4412	825	1	j.	j.	PROPN
ejpam-4412	825	2	inequal	inequal	PROPN
ejpam-4412	825	3	.	.	PUNCT
ejpam-4412	826	1	appl	appl	PROPN
ejpam-4412	826	2	.	.	PROPN
ejpam-4412	826	3	,	,	PUNCT
ejpam-4412	826	4	7:143–168	7:143–168	NOUN
ejpam-4412	826	5	,	,	PUNCT
ejpam-4412	826	6	2002	2002	NUM
ejpam-4412	826	7	.	.	PUNCT
ejpam-4412	827	1	[	[	X
ejpam-4412	827	2	34	34	NUM
ejpam-4412	827	3	]	]	X
ejpam-4412	827	4	c.	c.	PROPN
ejpam-4412	827	5	yang	yang	PROPN
ejpam-4412	827	6	and	and	CCONJ
ejpam-4412	827	7	j.	j.	PROPN
ejpam-4412	827	8	yuan	yuan	PROPN
ejpam-4412	827	9	.	.	PUNCT
ejpam-4412	828	1	on	on	ADP
ejpam-4412	828	2	class	class	NOUN
ejpam-4412	828	3	wf	wf	PROPN
ejpam-4412	828	4	(	(	PUNCT
ejpam-4412	828	5	p	p	X
ejpam-4412	828	6	,	,	PUNCT
ejpam-4412	828	7	r	r	NOUN
ejpam-4412	828	8	,	,	PUNCT
ejpam-4412	828	9	q	q	NOUN
ejpam-4412	828	10	)	)	PUNCT
ejpam-4412	828	11	operators	operator	NOUN
ejpam-4412	828	12	.	.	PUNCT
ejpam-4412	829	1	acta	acta	PROPN
ejpam-4412	829	2	.	.	PUNCT
ejpam-4412	830	1	math	math	NOUN
ejpam-4412	830	2	.	.	PUNCT
ejpam-4412	831	1	sci	sci	PROPN
ejpam-4412	831	2	.	.	PROPN
ejpam-4412	831	3	,	,	PUNCT
ejpam-4412	831	4	27:769–780	27:769–780	NUM
ejpam-4412	831	5	,	,	PUNCT
ejpam-4412	831	6	2007	2007	NUM
ejpam-4412	831	7	.	.	PUNCT
ejpam-4412	832	1	[	[	X
ejpam-4412	832	2	35	35	NUM
ejpam-4412	832	3	]	]	X
ejpam-4412	832	4	j.	j.	PROPN
ejpam-4412	832	5	yuan	yuan	PROPN
ejpam-4412	832	6	and	and	CCONJ
ejpam-4412	832	7	c.	c.	PROPN
ejpam-4412	832	8	yang	yang	PROPN
ejpam-4412	832	9	.	.	PUNCT
ejpam-4412	833	1	spectrum	spectrum	NOUN
ejpam-4412	833	2	of	of	ADP
ejpam-4412	833	3	class	class	NOUN
ejpam-4412	833	4	wf	wf	PROPN
ejpam-4412	833	5	(	(	PUNCT
ejpam-4412	833	6	p	p	X
ejpam-4412	833	7	,	,	PUNCT
ejpam-4412	833	8	r	r	NOUN
ejpam-4412	833	9	,	,	PUNCT
ejpam-4412	833	10	q	q	NOUN
ejpam-4412	833	11	)	)	PUNCT
ejpam-4412	833	12	operators	operator	NOUN
ejpam-4412	833	13	for	for	ADP
ejpam-4412	833	14	p+r	p+r	NUM
ejpam-4412	833	15	≤	≤	NUM
ejpam-4412	833	16	1	1	NUM
ejpam-4412	833	17	and	and	CCONJ
ejpam-4412	833	18	q	q	ADJ
ejpam-4412	833	19	≥	≥	NUM
ejpam-4412	833	20	1	1	NUM
ejpam-4412	833	21	.	.	PUNCT
ejpam-4412	833	22	acta	acta	PROPN
ejpam-4412	833	23	sci	sci	PROPN
ejpam-4412	833	24	.	.	PROPN
ejpam-4412	833	25	math	math	PROPN
ejpam-4412	833	26	.	.	PUNCT
ejpam-4412	834	1	(	(	PUNCT
ejpam-4412	834	2	szeged	szeged	PROPN
ejpam-4412	834	3	)	)	PUNCT
ejpam-4412	834	4	,	,	PUNCT
ejpam-4412	835	1	71:767–779	71:767–779	PROPN
ejpam-4412	835	2	,	,	PUNCT
ejpam-4412	835	3	2005	2005	NUM
ejpam-4412	835	4	.	.	PUNCT
ejpam-4412	836	1	[	[	X
ejpam-4412	836	2	36	36	NUM
ejpam-4412	836	3	]	]	X
ejpam-4412	836	4	j.	j.	PROPN
ejpam-4412	836	5	yuan	yuan	PROPN
ejpam-4412	836	6	and	and	CCONJ
ejpam-4412	836	7	c.	c.	PROPN
ejpam-4412	836	8	yang	yang	PROPN
ejpam-4412	836	9	.	.	PUNCT
ejpam-4412	837	1	powers	power	NOUN
ejpam-4412	837	2	of	of	ADP
ejpam-4412	837	3	class	class	NOUN
ejpam-4412	837	4	wf	wf	PROPN
ejpam-4412	837	5	(	(	PUNCT
ejpam-4412	837	6	p	p	X
ejpam-4412	837	7	,	,	PUNCT
ejpam-4412	837	8	r	r	NOUN
ejpam-4412	837	9	,	,	PUNCT
ejpam-4412	837	10	q	q	NOUN
ejpam-4412	837	11	)	)	PUNCT
ejpam-4412	837	12	operators	operator	NOUN
ejpam-4412	837	13	.	.	PUNCT
ejpam-4412	838	1	journal	journal	PROPN
ejpam-4412	838	2	inequalities	inequality	NOUN
ejpam-4412	838	3	in	in	ADP
ejpam-4412	838	4	pure	pure	ADJ
ejpam-4412	838	5	and	and	CCONJ
ejpam-4412	838	6	applied	applied	ADJ
ejpam-4412	838	7	math	math	NOUN
ejpam-4412	838	8	.	.	PUNCT
ejpam-4412	838	9	,	,	PUNCT
ejpam-4412	838	10	7(1):article	7(1):article	NUM
ejpam-4412	838	11	32	32	NUM
ejpam-4412	838	12	,	,	PUNCT
ejpam-4412	838	13	2006	2006	NUM
ejpam-4412	838	14	.	.	PUNCT